@@hugowong4436are you saying that because 4 and 9 are both square numbers? Because that’s not necessary for the geometric solution to work. It just makes the math easier to do, and acts as a little clue to help you find the geometric solution.
@@SmoMo_And it can also even work if there is a term bx added underneath the first square root (with |b| ≤ 2√4 ), and/or a term qy added underneath the second square root (with |q| ≤ 2√9 ).
The algebra is simplified considerably if you use implicit differentiation. x and y are trivially separable when setting f’ to zero, and you already know from x+y=12 that dy/dx=-1.
Wow, the geometry method is GREAT! I did not expect that. Even when the two triangles were drawn, I thought: Well, clever, but how are you going proceed from here? And then you did the mirroring in the horizontal axis.... flabber-gasted! 😮😊 Excellent!
Lovely geometric solution 😍 I learnt it as "the problem of the firefighter". As usual, thanks for sharing! With calculus you can also use the advanced technique of Lagrange multipliers (multivariable calculus), that leads to a shorter solution: we want to minimize f(x,y)=√(4+x²)+√(9+y²) under the restriction x+y=12. Let's call g(x,y)=x+y-12 and consider the function L(x,y,λ) = f(x,y)+λg(x,y) = √(4+x²)+√(9+y²)+λ(x+y-12). Then to find a local max/min we impose ∂L/∂x=0, ∂L/∂y=0, ∂L/∂λ=0. We obtain a quite simple system of equations from which we find the minimum x=24/5, y=36/5.
And from the geometry, it is easy to compute x (and y) using the fact that the two triangles are similar, so y/x = 3/2. This together with x+y=12 recovers x = 24/5. Brilliant!
Since i was 4 or 5, i was fascinated about maths and loved it so much. I would practise problems and learn quickly about maths. But when things got hard in maths, you were always there which gave me motivation to get to where i am right now, which is the reason i love maths.
If you have studied ray optics you might use fermats rule to find the minimum distance after reflection, because light will take the shortest path and you can put angle of incidence=angle of reflection and get the result this way instead of flipping the green triangle
Having learnt the optimization rule about 2 weeks ago, the moment you created the two triangles and animated the point's shift at 4:41, it all clicked in.
I did something similar, I constructed two triangles, one on the right an over the other one and you can see that the minimum distance creates another triangle.
2 original triangles are already similar (angle-angle), so x/2 = y/3 = => x = 24/5 and y = 36/5. Let u = x/2 and v = y/3. then u = v = 12/5. f(x) = 2*sqrt(1 + u^1) + 3*sqrt(1 + v^2) = 5*sqrt(1 + u^2) = 5*sqrt(5^2/5^2 + 12^2/5^2) = sqrt(5^2 + 12^2) = 13.
Well I got the answer but I had to test all the possibilities. The geometry approach was amazing when you said "we can use geometry to solve it" I immediately stop the video and tried to figure it out by myself and I was able to draw the x+y=12 and those triangles but got stuck there xD.
First is the distance from O(0, 0) to A(2, x) and second is the distance from A(2, x) to B(5, x+y). The straight path from O(0, 0)to B(5, 12) is the required minimum distance = 13.
Amazing video as always!! I found another solution with Cauchy-Schwarz inequality (it's actually the geometry solution but without the geometry lol), let t=√(4+x²)+√(9+y²) so t²=13+x²+y²+2√((4+x²)(9+y²)). From the Cauchy-Schwarz inequality we get √((4+x²)(9+y²))=√((2²+x²)(3²+y²))≥6+xy, so t²≥13+x²+y²+2(6+xy)=25+(x+y)²=25+144=169 and because t is positive we get t minimal is 13! we also know that the Cauchy-Schwarz inequality becomes equal only if the vectors (2,x),(3,y) are linearly dependent, so 2/3=x/y and x+y=12 which give us x=24/5 and y=36/5 for the minimal case.
@@8DJYashya,that's coincidence it took half an hour to use the cauchy schwarz's and then..solve It's quite impressive that the expression changes into (x+y)^2
I have just found a very good generalization of the problem : let ai in IR+ be n constants. Let xi in IR+ such that Σ xi = q Let S = Σ √(xi²+ai²) The minimum value of S is √(q²+(Σ ai)²)
With calculus it's much easier using conditional extremes. You take the function φ = √(4+x²) + √(9+y²) - λ(x + y -12). You derive wrt x, y and λ and get a much simpler system for the three variables. The system asks for the change of variable x = 2 tg(u), y = 3 tg(v). It is immediate that u = v and then the solutions are 24/5 and 36/5.
The geometric solution is really neat. However the calculus simplifies a lot if you just differentiate the sum of square roots wrt x using implicit differentiation and compare it to -1 (the slope of x+y=12 line). This will give the relationship 3x=2y, which then is trivially solved.
@@Ideophagous Differentiation before replacing y = 12-x ; so you're leaving both x and y in the expression to differentiate, and then try to derive a relation between x and y. f = √(4 + x²) + √(9 + y²) ... d is the differentiation operation ... df = x*dx/√(4 + x²) + y*dy/√(9 + y²) df = x*dx/√(4 + x²) + y/√(9 + y²) * (dy/dx) * dx ... x + y = 12, so dx + dy = 0 , hence dy/dx = -1 ... df = x*dx/√(4 + x²) + y/√(9 + y²) * (-1) * dx df/dx = x/√(4 + x²) - y/√(9 + y²) From there on, by setting df/dx = 0, we can find a relation between x and y where f has its (local/global) extremum.
I used a similar method, but working off the ratios of the sides of the triangles being equal (2/x=3/y), meaning the hypotenuses would be at the same angle. This forms a straight line when the two triangles are joined in the same orientation.
You can also use multivariable calculus. To minimize f(x,y)=sqrt(4+x^2)+sqrt(9+y^2) with g(x,y)=x+y-12=0 you have to solve the system d(f+λg)=0 which, when combined with g = 0, gives a solution for λ. Then x and y are calculated as they are connected with λ and these values will minimize f.
Damn. I just had Operation Research 2 last semester and optimizing nonlinear functions was part of the syllabus. But I've already forgotten everything.
After watching the video, the solution seems much simpler than what I had in my mind. We used to calculate some matrix to determine if the function has a maximum in the df = 0 solution or not. But can't remember specifics so I'm not sure why here that step is not needed.
for the first method, you must prove the function is continuous. After that proving derivation is 0 in some x, you proved in the point x there is minimum OR maximum. or none of them. You must check whether the function is increasing or decreasing on left and right of the value x. For example for function f(x) = x^2 there is minimum for x= 0, for f(x)= - x^2 there is maximum for x=0 and for function f(x) = x^3 there is not a minimum, nor maximum for x=0.
I immediately thought it was a “double Gougu” puzzle, but I wasn’t sure how to set it up. At one point I even thought it would involve two lines that would give the answer when perpendicular.
i ignored the without calculus and solved it using calculus, although impilicit differentiation is so much easier, which is what i did. easier calculus method. people who know calculus would find this the most obviousl and intuitive way to go. geometry method is hard to think of for me at least. it works out great, but the thought process to think of it when u know calculus doesnt really pop up.
القيمة التي تنعدم عندها المشتقة الاولى هي قيمة حدية و لا نعرف اذا كانت قيمة حدية صغرى او قيمة حدية عظمى و يجب الذهاب الى المشتقة الثانية. في الطريقة الهندسية اعتمدت على شرط ان x>0 و y>0 و هذا ليس من معطيات المسألة.
For those who’ve heard of it, would the method of Lagrange multipliers work? Specifically, by setting f(x,y) = √(4+x²) + √(9+y²) and apply the constraint x + y - 12 = 0.
we want to minimize f(x,y)=√(4+x²)+√(9+y²) under the restriction x+y=12. Let's call g(x,y)=x+y-12 and consider the function L(x,y,λ) = f(x,y)+λg(x,y) = √(4+x²)+√(9+y²)+λ(x+y-12). Then to find a local max/min we impose ∂L/∂x=0, ∂L/∂y=0, ∂L/∂λ=0. We obtain a quite simple system of equations from which we find the minimum x=24/5, y=36/5.
I just don't understand why for the geometric proof, instead of using Pythagoras, you don't just check the distance between A and B by using the cartesian coordinates: the distance between A(0,2) and B(12,-3) is 13 obviously 🤷♂️ But great video, as usual.
When I read the problem i instantly thought of some geometrical solution for the problem, Maxima Minima is standard way of solving, but i thought using circle we can get it but not able to get using right triangles
the first part could have been done much easier and more elegant using Lagrange multiplier and the Geometry method is just demonstration of the famous Fermat principle in optics
Your geometric proofs are pretty amazing, and this is a good case in point! However, I'd like to point out an obvious fact you glossed over, namely that the shortest distance between two points in a plane is a straight line. Although this is intuitively obvious, the proof involves calculus of variations, which is much harder than first-year calculus!
The fact that a straight line is the shortest path can be proven by repeatedly applying triangle inequality to the approximations of the curve. In this particular case, the curve is made of 2 straight lines, so it's simply one application of triangle inequality. Calculus of variation is straight up overkill for the problem.
It might be easier but to be perfectly honest I find the calculus solution to be so much more entertaining because I just love numbers way too much haha
Hello presh tawalkar I was searching an algebra problem from your channel I found one problem there you were asking x *5 + y*5 + z*5 and x*6 +y*6+z*6 I got both answer using algebra answer are 6 and 103/12 are answers correct
Isn't that the video with the problem Given x + y + z = 1 x² + y² + z² = 2 x³ + y³ + z³ = 3 Calculate x⁵+y⁵+z⁵ . If I remember correctly, if we'd label xⁿ+yⁿ+zⁿ as p(n), then we could eventually derive the following recurrence relation for p(n): p(n+3) = p(n+2) + (1/2)*p(n+1) + (1/6)*p(n) From there, we could calculate p(4) = 3 + (1/2)*2 + (1/6)*1 = 25/6 p(5) = 25/6 + (1/2)*3 + (1/6)*2 = 25/6 + 9/6 + 2/6 = 36/6 = *6* p(6) = 6 + (1/2)*25/6 + (1/6)*3 = 72/12 + 25/12 + 6/12 = *103/12* So if I'm right, then your answers are also correct. However, Presh's video delved much deeper into generalizing the problem, and addressed Girard-Newton identities (which became rather challenging to follow and understand, but which is really worth it).
Yes, but how would you have known its gonna give you a way to solve the problem after all? Calculus just feels safer to use and one derivative and some multiplication is not that bad
yea he just showing the cool solution. obviously if u see this problem u shouldnt try to think of a creative solution and just use calculus if it isnt super complicated. here the calculus was easy. u can also implicitly differentiate which makes it much more simpler as you already know dy/dx = -1.
Question: when you say that quadratic is "easily" factorisable - is there a method other than churning through the factors of 2880 until you get the right combination 96x = -24x +120x ?
I don't know how Presh did it, and I don't know if he really found it "easy" (maybe it was meant to be ironic, as a set-up that pays out later when the geometry method turns out to be so much more easy-going, as it doesn't juggle with that many numbers). It doesn't look easy to me either, unless to someone who has memorised the multiplication table of 24. Personally I would have solved it by the method of completing the square. But upon your question, I had a look at the equation and noticed something. 5x² + 96x - 576 = 0 (eq. 1) 576 is 24 squared, and 96 is a multiple of 24. So the proper factoring of 2880 = 5*576 is probably going to preserve a divisor of 24 in both factors, and the summands to 96 must be proportioned according to a factoring of -5 = (-1)*(+5) , hence we "quickly" arrive at 96x = (-1)*(24x) + (+5)*(24x) . Another way to see this, is by replacing 24 by a variable, say y. The equation becomes 5x² + 4xy - y² = 0 (eq. 2) This looks much easier to factorise: just as we remember from high school, we have to look for factors of -5 that add to 4 . In fact, this is the same problem as factoring 5u² + 4u - 1 = 0 (eq. 3) (It _is_ indeed the same problem; just divide (eq. 2) by y² , and then substitute x/y = u , and we get (eq. 3). )
of course, The way anybody solves a problem always seems simpler to them. Before anybody would be too impressed with your simpler way, you would first have to explain to them why Lagrange multiplier even results in a max/min The geometric approach in this video did not require any understanding of results that might be hard to prove, only some rudimentary geometry.
The geometry is a neat explanation but it’s wrong. The angle ACx is not the same as the angle BCy, which means that ACB is not a straight line (it just looks like that; but it’s not correct). What you’re actually trying to do is : given x+y =c, minimise x^2 + y^2. This happens when x=y. So the correct answer of the minimum value of your equation is sqrt(40) + sqrt(45) ~13.03.
I did it the same way , but i framed the problem differently . I considered a number line and the values are distances on cordinate planes.. got result intuitively after that
Hmm, you go as far as saying the sqroot(4+sq(x)) + sqroot(9+sq(y)) is 13, in the first solution you mention that x is 24/5 and the initial equation is x+y=12, but you never actually give the value of y. So... never actually gives the answer of both x and y for the answer. Yes it's only a single step away, but still it makes it feel incomplete.
An extraneous solution is a solution that is found at the end of the calculation, but that isn't a solution of the original equation. For example, suppose we're solving the equation x = √(x+6) Squaring both sides gives us x² = x + 6 x² - x - 6 = 0 (x - 3)(x + 2) = 0 x = 3 OR x = -2 From these two "solutions", only x=3 is a solution of the original equation. The other solution, x = -2 , does not solve the original equation and hence is an extraneous solution. It was introduced by the step of squaring both sides. Notice that x = -2 is a solution of the equation x = -√(x+6) which, when squaring, results in the same quadratic equation, but obviously isn't the same equation as the original equation x = +√(x+6) .
I used multivariable calculus to solve for x. Slightly easier calculations as I was able to get x=24/5 in my head. Still not as nice as the geometry solution.
The geometric solution is the reason for which I love Maths.
I think the way is converse. This question is specifically designed for the solution. The ans seems clever because it is designed for that
@@hugowong4436i was about to type the same thing but in a less articulated way
@@hugowong4436are you saying that because 4 and 9 are both square numbers?
Because that’s not necessary for the geometric solution to work. It just makes the math easier to do, and acts as a little clue to help you find the geometric solution.
@@SmoMo_And it can also even work if there is a term bx added underneath the first square root (with |b| ≤ 2√4 ), and/or a term qy added underneath the second square root (with |q| ≤ 2√9 ).
Very beautiful. I didn’t even think that it is possible to solve this in geometric way.Thanks for teaching us such great things
The algebra is simplified considerably if you use implicit differentiation. x and y are trivially separable when setting f’ to zero, and you already know from x+y=12 that dy/dx=-1.
Your approach is equivalent to the method using Lagrange Multiplicators 🙂.
Wow, the geometry method is GREAT! I did not expect that. Even when the two triangles were drawn, I thought: Well, clever, but how are you going proceed from here? And then you did the mirroring in the horizontal axis.... flabber-gasted! 😮😊
Excellent!
Geometric approach is wonderful
Lovely geometric solution 😍 I learnt it as "the problem of the firefighter". As usual, thanks for sharing!
With calculus you can also use the advanced technique of Lagrange multipliers (multivariable calculus), that leads to a shorter solution: we want to minimize f(x,y)=√(4+x²)+√(9+y²) under the restriction x+y=12. Let's call g(x,y)=x+y-12 and consider the function L(x,y,λ) = f(x,y)+λg(x,y) = √(4+x²)+√(9+y²)+λ(x+y-12). Then to find a local max/min we impose ∂L/∂x=0, ∂L/∂y=0, ∂L/∂λ=0. We obtain a quite simple system of equations from which we find the minimum x=24/5, y=36/5.
Wonderful, for some reason Lagrange multipliers was always part of my economics classes instead of mathematics.
❤ wonderful
How did u recieve this vid 2days ago??😮
@@Kshitij.with.nature-channel he only sent to me and Nestor, it was private
@@jgfla2nahh u tripin
And from the geometry, it is easy to compute x (and y) using the fact that the two triangles are similar, so y/x = 3/2. This together with x+y=12 recovers x = 24/5. Brilliant!
the geometric solution is absolutely mindblowing
Since i was 4 or 5, i was fascinated about maths and loved it so much. I would practise problems and learn quickly about maths. But when things got hard in maths, you were always there which gave me motivation to get to where i am right now, which is the reason i love maths.
Amazing Solution,
Those expressions made me think of the circle's equation and I couldn't think of Geometrical Solution
yea i thought that aswell
Geometry approach was really elegantly explained...thanks...👍🏻
If you have studied ray optics you might use fermats rule to find the minimum distance after reflection, because light will take the shortest path and you can put angle of incidence=angle of reflection and get the result this way instead of flipping the green triangle
Yea but i don't think it's good to combine physics in a purely mathematical problem
@@epikherolol8189 I know you aren't supposed to do that in an exam but it just clicked into my mind that this is also a way to think about it
@@epikherolol8189physics is just applied math
That was my solution too lol.
Heart filled with joy to watch your geometry solution .God bless you .❤❤❤
That's a very elegant geometric solution.
Having learnt the optimization rule about 2 weeks ago, the moment you created the two triangles and animated the point's shift at 4:41, it all clicked in.
I did something similar, I constructed two triangles, one on the right an over the other one and you can see that the minimum distance creates another triangle.
MAGNIFICENT GEOMETRY SOLUTION!!!
Great question and solution.
2 original triangles are already similar (angle-angle), so x/2 = y/3 = => x = 24/5 and y = 36/5. Let u = x/2 and v = y/3. then u = v = 12/5.
f(x) = 2*sqrt(1 + u^1) + 3*sqrt(1 + v^2) = 5*sqrt(1 + u^2) = 5*sqrt(5^2/5^2 + 12^2/5^2) = sqrt(5^2 + 12^2) = 13.
Amazing❤
You always make my day
Love from india❤
Can be done from properties of AM & GM, a+b/2
Well I got the answer but I had to test all the possibilities. The geometry approach was amazing when you said "we can use geometry to solve it" I immediately stop the video and tried to figure it out by myself and I was able to draw the x+y=12 and those triangles but got stuck there xD.
geometry solution was amazing.. love it
First is the distance from O(0, 0) to A(2, x) and second is the distance from A(2, x) to B(5, x+y). The straight path from O(0, 0)to B(5, 12) is the required minimum distance = 13.
great solutions thanks for the video.
Amazing video as always!! I found another solution with Cauchy-Schwarz inequality (it's actually the geometry solution but without the geometry lol), let t=√(4+x²)+√(9+y²) so t²=13+x²+y²+2√((4+x²)(9+y²)). From the Cauchy-Schwarz inequality we get √((4+x²)(9+y²))=√((2²+x²)(3²+y²))≥6+xy, so t²≥13+x²+y²+2(6+xy)=25+(x+y)²=25+144=169 and because t is positive we get t minimal is 13! we also know that the Cauchy-Schwarz inequality becomes equal only if the vectors (2,x),(3,y) are linearly dependent, so 2/3=x/y and x+y=12 which give us x=24/5 and y=36/5 for the minimal case.
I also used inequalities too.
Solving an equality by inequality is amazing
On the other hand,Calculus is feeling jealous.
we both have exactly same approach!
@@8DJYashya,that's coincidence
it took half an hour to use the cauchy schwarz's and then..solve
It's quite impressive that the expression changes into (x+y)^2
Loved the Geometrical Solution !❤
I have just found a very good generalization of the problem :
let ai in IR+ be n constants.
Let xi in IR+ such that Σ xi = q
Let S = Σ √(xi²+ai²)
The minimum value of S is √(q²+(Σ ai)²)
That's very satisfying.
Geometric solution is great, calculus is the reason why I love mathematics.
The geometric solution is damn gorgeous.
With calculus it's much easier using conditional extremes. You take the function φ = √(4+x²) + √(9+y²) - λ(x + y -12). You derive wrt x, y and λ and get a much simpler system for the three variables. The system asks for the change of variable x = 2 tg(u), y = 3 tg(v). It is immediate that u = v and then the solutions are 24/5 and 36/5.
This was a great trick.
Good Job
It is the geometric terminology that may get some, but the term should and could equal 12 too.
The geometric solution is really neat.
However the calculus simplifies a lot if you just differentiate the sum of square roots wrt x using implicit differentiation and compare it to -1 (the slope of x+y=12 line). This will give the relationship 3x=2y, which then is trivially solved.
What do you mean by implicit differentiation?
@@Ideophagous Differentiation before replacing y = 12-x ; so you're leaving both x and y in the expression to differentiate, and then try to derive a relation between x and y.
f = √(4 + x²) + √(9 + y²)
... d is the differentiation operation ...
df = x*dx/√(4 + x²) + y*dy/√(9 + y²)
df = x*dx/√(4 + x²) + y/√(9 + y²) * (dy/dx) * dx
... x + y = 12, so dx + dy = 0 , hence dy/dx = -1 ...
df = x*dx/√(4 + x²) + y/√(9 + y²) * (-1) * dx
df/dx = x/√(4 + x²) - y/√(9 + y²)
From there on, by setting df/dx = 0, we can find a relation between x and y where f has its (local/global) extremum.
This was awesome!
I used a similar method, but working off the ratios of the sides of the triangles being equal (2/x=3/y), meaning the hypotenuses would be at the same angle. This forms a straight line when the two triangles are joined in the same orientation.
You can also use multivariable calculus. To minimize f(x,y)=sqrt(4+x^2)+sqrt(9+y^2) with g(x,y)=x+y-12=0 you have to solve the system d(f+λg)=0 which, when combined with g = 0, gives a solution for λ. Then x and y are calculated as they are connected with λ and these values will minimize f.
No need to use Lagrange multipliers for this problem and we have to do that without using calculus
Great video , what's the name of the music at the end ?
Damn. I just had Operation Research 2 last semester and optimizing nonlinear functions was part of the syllabus. But I've already forgotten everything.
After watching the video, the solution seems much simpler than what I had in my mind. We used to calculate some matrix to determine if the function has a maximum in the df = 0 solution or not. But can't remember specifics so I'm not sure why here that step is not needed.
for the first method, you must prove the function is continuous. After that proving derivation is 0 in some x, you proved in the point x there is minimum OR maximum. or none of them. You must check whether the function is increasing or decreasing on left and right of the value x. For example for function f(x) = x^2 there is minimum for x= 0, for f(x)= - x^2 there is maximum for x=0 and for function f(x) = x^3 there is not a minimum, nor maximum for x=0.
Thanks, nice one..👍
I immediately thought it was a “double Gougu” puzzle, but I wasn’t sure how to set it up. At one point I even thought it would involve two lines that would give the answer when perpendicular.
Incidentally, I solved this problem using calculus, but I also used a Lagrange multiplier, which makes the calculation much less messy!
U can also use lagrange multiplier concept to solve optimization problem
I honestly felt like geometry was the more realistic way of answering this problem before the calculus way
i ignored the without calculus and solved it using calculus, although impilicit differentiation is so much easier, which is what i did. easier calculus method. people who know calculus would find this the most obviousl and intuitive way to go. geometry method is hard to think of for me at least. it works out great, but the thought process to think of it when u know calculus doesnt really pop up.
By differentiation is easier
Put y=x-12
And continue 😊
Geometry was ❤from INDIA 🎉
The geometrical reasoning behind this problem made the Calculus approach look like using a flamethrower to kill a mosquito.
It is easy to do this using just the triangle inequality for vectors:
13=|(12, 5)|=|(x+y,5)|=|(x,2)+(y,3)|≤|(x,2)|+|(y,3)|
القيمة التي تنعدم عندها المشتقة الاولى هي قيمة حدية و لا نعرف اذا كانت قيمة حدية صغرى او قيمة حدية عظمى و يجب الذهاب الى المشتقة الثانية.
في الطريقة الهندسية اعتمدت على شرط ان x>0 و y>0 و هذا ليس من معطيات المسألة.
Calculus is more general and easier. A geometric solution when revealed is easy to appreciate but not easy to derive and may not always exist.
The point on the line x+y=12 closest to the origin. The objective function increases away from the origin.
Incredible!
The geometric way is very elegant.
Nice pair of hypotenuses, said my Hippopotamuses when they saw this awesome trick.
For those who’ve heard of it, would the method of Lagrange multipliers work? Specifically, by setting f(x,y) = √(4+x²) + √(9+y²) and apply the constraint x + y - 12 = 0.
we want to minimize f(x,y)=√(4+x²)+√(9+y²) under the restriction x+y=12. Let's call g(x,y)=x+y-12 and consider the function L(x,y,λ) = f(x,y)+λg(x,y) = √(4+x²)+√(9+y²)+λ(x+y-12). Then to find a local max/min we impose ∂L/∂x=0, ∂L/∂y=0, ∂L/∂λ=0. We obtain a quite simple system of equations from which we find the minimum x=24/5, y=36/5.
I just don't understand why for the geometric proof, instead of using Pythagoras, you don't just check the distance between A and B by using the cartesian coordinates: the distance between A(0,2) and B(12,-3) is 13 obviously 🤷♂️
But great video, as usual.
When I read the problem i instantly thought of some geometrical solution for the problem, Maxima Minima is standard way of solving, but i thought using circle we can get it but not able to get using right triangles
I tried to somehow use AM-GM inequality on this but I never knew this can be solved geometrically.
the first part could have been done much easier and more elegant using Lagrange multiplier and the Geometry method is just demonstration of the famous Fermat principle in optics
You just demonstrated the law of light reflection
Your geometric proofs are pretty amazing, and this is a good case in point! However, I'd like to point out an obvious fact you glossed over, namely that the shortest distance between two points in a plane is a straight line. Although this is intuitively obvious, the proof involves calculus of variations, which is much harder than first-year calculus!
The fact that a straight line is the shortest path can be proven by repeatedly applying triangle inequality to the approximations of the curve. In this particular case, the curve is made of 2 straight lines, so it's simply one application of triangle inequality.
Calculus of variation is straight up overkill for the problem.
@@Noname-67 OK, fair enough! Thank you for clarifying this.
1:33 square of 12 - x squared... :)
Amazing trick......
It might be easier but to be perfectly honest I find the calculus solution to be so much more entertaining because I just love numbers way too much haha
Nicolas Bourbaki would be proud of you
0:45 "So let's get started with calculus," said no one ever.
Out of the box thinking 🤔🧐
Why not to use the Cashy Swarz?
Let z1=x+2i and z2= y+3i, and then apply mod z1+ mod z2 》mod(z1+z2)
Can you solve it with polar co-ordinates?
Really cool
Why is f’(-24) = - 24 / sqrt (145) ?
It should be 0.
Hello presh tawalkar I was searching an algebra problem from your channel I found one problem there you were asking x *5 + y*5 + z*5 and x*6 +y*6+z*6 I got both answer using algebra answer are 6 and 103/12 are answers correct
Isn't that the video with the problem
Given
x + y + z = 1
x² + y² + z² = 2
x³ + y³ + z³ = 3
Calculate x⁵+y⁵+z⁵ .
If I remember correctly, if we'd label xⁿ+yⁿ+zⁿ as p(n), then we could eventually derive the following recurrence relation for p(n):
p(n+3) = p(n+2) + (1/2)*p(n+1) + (1/6)*p(n)
From there, we could calculate
p(4) = 3 + (1/2)*2 + (1/6)*1 = 25/6
p(5) = 25/6 + (1/2)*3 + (1/6)*2 = 25/6 + 9/6 + 2/6 = 36/6 = *6*
p(6) = 6 + (1/2)*25/6 + (1/6)*3 = 72/12 + 25/12 + 6/12 = *103/12*
So if I'm right, then your answers are also correct.
However, Presh's video delved much deeper into generalizing the problem, and addressed Girard-Newton identities (which became rather challenging to follow and understand, but which is really worth it).
Do you edit your voice or is this your actual voice?
Yes, but how would you have known its gonna give you a way to solve the problem after all? Calculus just feels safer to use and one derivative and some multiplication is not that bad
yea he just showing the cool solution. obviously if u see this problem u shouldnt try to think of a creative solution and just use calculus if it isnt super complicated. here the calculus was easy. u can also implicitly differentiate which makes it much more simpler as you already know dy/dx = -1.
Question: when you say that quadratic is "easily" factorisable - is there a method other than churning through the factors of 2880 until you get the right combination 96x = -24x +120x ?
I don't know how Presh did it, and I don't know if he really found it "easy" (maybe it was meant to be ironic, as a set-up that pays out later when the geometry method turns out to be so much more easy-going, as it doesn't juggle with that many numbers). It doesn't look easy to me either, unless to someone who has memorised the multiplication table of 24. Personally I would have solved it by the method of completing the square.
But upon your question, I had a look at the equation and noticed something.
5x² + 96x - 576 = 0 (eq. 1)
576 is 24 squared, and 96 is a multiple of 24. So the proper factoring of 2880 = 5*576 is probably going to preserve a divisor of 24 in both factors, and the summands to 96 must be proportioned according to a factoring of -5 = (-1)*(+5) , hence we "quickly" arrive at 96x = (-1)*(24x) + (+5)*(24x) .
Another way to see this, is by replacing 24 by a variable, say y. The equation becomes
5x² + 4xy - y² = 0 (eq. 2)
This looks much easier to factorise: just as we remember from high school, we have to look for factors of -5 that add to 4 . In fact, this is the same problem as factoring
5u² + 4u - 1 = 0 (eq. 3)
(It _is_ indeed the same problem; just divide (eq. 2) by y² , and then substitute x/y = u , and we get (eq. 3). )
Using complex no with (x+y)i=12i and taking modulus of both gives the same result like geometry
at the geometric solution I tought you will go for the final solution of: x/12 = 2/5 -> x = 24/5
I sketched a solution using Lagrange multipliers, and it seemed to be a lot simpler.
of course,
The way anybody solves a problem always seems simpler to them.
Before anybody would be too impressed with your simpler way, you would first have to explain to them why Lagrange multiplier even results in a max/min
The geometric approach in this video did not require any understanding of results that might be hard to prove, only some rudimentary geometry.
The geometry is a neat explanation but it’s wrong. The angle ACx is not the same as the angle BCy, which means that ACB is not a straight line (it just looks like that; but it’s not correct).
What you’re actually trying to do is : given x+y =c, minimise x^2 + y^2. This happens when x=y. So the correct answer of the minimum value of your equation is sqrt(40) + sqrt(45) ~13.03.
Can you use Lagrange multiplier for this
Read the title. The idea is that you can do this without calculus.
great
I did it the same way , but i framed the problem differently . I considered a number line and the values are distances on cordinate planes.. got result intuitively after that
Excuse me sir but u didn't tell the values of x and y? Btw I love ur vids, this was a great and very helpful question😅
x = 24/5 and y = 12 - x.
Guys how do we quickly factorisation that quadratic, at 2:36
I love math because it has no exception
If there was bound for x and y i could solve using coordinate geometry 🥰
amazing
That little maneuver will cost us meme goes here
Geometric beautiful
I can do it answer is 13
By Pythagoras
Hmm, you go as far as saying the sqroot(4+sq(x)) + sqroot(9+sq(y)) is 13, in the first solution you mention that x is 24/5 and the initial equation is x+y=12, but you never actually give the value of y. So... never actually gives the answer of both x and y for the answer. Yes it's only a single step away, but still it makes it feel incomplete.
Suppose we take x²+4 as equal to y²+9.
This vid is uploaded 13mins ago, how did 12 of u members commented many hours ago????
I think this can be done somehow by using the square root of a negative number.
Sometimes I wish I could know maths at close as you
Does C++ qualify as "without calculus"?
What is a Extraneous solution ?
An extraneous solution is a solution that is found at the end of the calculation, but that isn't a solution of the original equation.
For example, suppose we're solving the equation
x = √(x+6)
Squaring both sides gives us
x² = x + 6
x² - x - 6 = 0
(x - 3)(x + 2) = 0
x = 3 OR x = -2
From these two "solutions", only x=3 is a solution of the original equation. The other solution, x = -2 , does not solve the original equation and hence is an extraneous solution. It was introduced by the step of squaring both sides. Notice that x = -2 is a solution of the equation
x = -√(x+6)
which, when squaring, results in the same quadratic equation, but obviously isn't the same equation as the original equation x = +√(x+6) .
I used multivariable calculus to solve for x. Slightly easier calculations as I was able to get x=24/5 in my head. Still not as nice as the geometry solution.