If I look at the given equation for about 30 seconds I see the solution and do not need any substitutions or Lambert w functions and 3 sheets of paper filled with calculations.
For 27^x + x = 0 Inspection reveals 1. (a) if x = 0, 1 + 0 > 0, If x > 0, 27^x + x > 0, by elimination x is negative. (b) if x = -1, 1/27 -1 < 0, x > -1. x is a negative fraction with an absolute value less than 1. 2. Factoring 27 gives 3^3 or 3*3*3. 3. (3^3)^x = - x by subtracting x from both sides of 27^x + x = 0. 4. x is a negative fraction giving ( 1/((3)^(3*-x))) = -x. 5. The multiple of (3*--x) is an integer, an irrational value would make 27^x and x, irrational. 6. The only value of x giving an integer value of (3* -x) is -1/3. ( - - is + ) 7. Insert x = -1/3 into 27^x + x = 0 gives (27^(-1/3) = 1/3 and x = (-1/3)) or (1/3) + (-1/3) = 0 Giving an answer of x = -1/3 : no evidence of solution by inspection among Oxford entrants.
27 is a perfect cube, so it makes sense to take the cube root (x =1/3), but since 27^x is always positive, x must be negative, so I tried x = -1/3, which worked out as a solution. While your method is much more rigorous, in an entrance exam it is more useful to be good at spotting answers quickly and leaving more time to work on harder questions. I do think the methods in the top comments are more reliable than my guesses though, so I’d use them!
27^x = -x (- x )^ (1/x) = 27 (-1/x)^ (-1/x) = 3^ 3 -1/x= 3 x= -1/3 Now derivate of 27^x+x is 27^x log 27 +1 which is always positive Hence function is increasing and only solution is x= -1/3
you must show that there is only one solution. 27^x = -x , LHS is strictly increasing , RHS is strictly decreasing => there is only one solution => x = -1/3.
27^x=-x 3^3x=-x LHS is positive & RHS is negative Put value of x in negative Check Put x=(-1/3) 3*(-1/3)=-1 We can see that, LHS= RHS So, x=-1/3 is answer
If I look at the given equation for about 30 seconds I see the solution and do not need any substitutions or Lambert w functions and 3 sheets of paper filled with calculations.
Excellent exercise! Thank you!
You're so welcome!🥰✅💕🙏
What!
Another good start to the morning, more math fun!
It sure is! Enjoy 👏👏👏
For 27^x + x = 0
Inspection reveals
1. (a) if x = 0, 1 + 0 > 0, If x > 0, 27^x + x > 0, by elimination x is negative.
(b) if x = -1, 1/27 -1 < 0, x > -1.
x is a negative fraction with an absolute value less than 1.
2. Factoring 27 gives 3^3 or 3*3*3.
3. (3^3)^x = - x by subtracting x from both sides of 27^x + x = 0.
4. x is a negative fraction giving ( 1/((3)^(3*-x))) = -x.
5. The multiple of (3*--x) is an integer, an irrational value would make 27^x and x, irrational.
6. The only value of x giving an integer value of (3* -x) is -1/3. ( - - is + )
7. Insert x = -1/3 into 27^x + x = 0 gives (27^(-1/3) = 1/3 and x = (-1/3)) or (1/3) + (-1/3) = 0
Giving an answer of x = -1/3 : no evidence of solution by inspection among Oxford entrants.
u^u=3^3 => u =3 ìf you can prove that f(x)=x^x is monotonic function.
27 is a perfect cube, so it makes sense to take the cube root (x =1/3), but since 27^x is always positive, x must be negative, so I tried x = -1/3, which worked out as a solution.
While your method is much more rigorous, in an entrance exam it is more useful to be good at spotting answers quickly and leaving more time to work on harder questions. I do think the methods in the top comments are more reliable than my guesses though, so I’d use them!
Oxford University Entrance Exam: 27ˣ + x = 0; x =?
x ≠ 0; x = - 27ˣ, x¹⸍ˣ = (- 27ˣ)¹⸍ˣ = - 27 = (- 3)³
x¹⸍ˣ = (- 3)⁽⁻¹⁾⁽⁻³⁾ = (- 3⁻¹)⁽⁻³⁾ = (- 1/3)¹⸍⁽⁻¹⸍³⁾; x = - 1/3
Answer check:
x = - 1/3: 27ˣ + x = 3³⁽⁻¹⸍³⁾ + (- 1/3) = 3⁻¹ - 3⁻¹ = 0; Confirmed
Final answer:
x = - 1/3
You are wrong!
You're mistaken, like the whole earthy people.
27^x = -x
(- x )^ (1/x) = 27
(-1/x)^ (-1/x) = 3^ 3
-1/x= 3
x= -1/3
Now derivate of 27^x+x is
27^x log 27 +1 which is always positive
Hence function is increasing and only solution is x= -1/3
It took 4 simple lines of algebra to get the W-function solution.
Engineering Mathematics is Interesting.
A lot of pure maths appears to be Solomon's Vanity.😮😮. But people are blind 🦮.
That's it!🙏🙏🙏
X=-1/3
27^x = 1/27^1/3
=1/3
27^x + x = 0
you must show that there is only one solution.
27^x = -x , LHS is strictly increasing , RHS is strictly decreasing
=> there is only one solution => x = -1/3.
27^x=-x
3^3x=-x
LHS is positive & RHS is negative
Put value of x in negative
Check Put x=(-1/3)
3*(-1/3)=-1
We can see that, LHS= RHS
So, x=-1/3 is answer
-1/3
Took me like 5 seconds
How, and I count the time!
Hello,
Oxford University Fails Entrance Exam Because
-1÷3=-3.33333333333e⁻¹ limit ➟ ∞ digits
➪ 27ˣ + x = 27⁻³.³³³³³³³³³³³ᵉ⁻¹ +( -3.33333333333e⁻¹) = 1.0⁻¹² ≠ 0.
Thank you for trying.