P(x)=2x^5+8x-7 P'(x)=10x^4+8>0 for any x so by the consequence of Theorem of Lagrange for the interval (-inf,inf) P(x) is strictly increasing because P(x)>0. Because lim (x->-inf) P(x)=-inf and lim (x->inf) P(x) -> inf by IVT there exists a point where P(x)=0 I was expecting P'(x) to have roots as this would involve T Rolle and be more interesting
neat - also follows immediately from Descartes's rule of signs
I never recall the exact statement of Descartes’s rule, even though I’ve used it in my own work in the past!
P(x)=2x^5+8x-7
P'(x)=10x^4+8>0 for any x so by the consequence of Theorem of Lagrange for the interval (-inf,inf) P(x) is strictly increasing because P(x)>0.
Because lim (x->-inf) P(x)=-inf and lim (x->inf) P(x) -> inf by IVT there exists a point where P(x)=0
I was expecting P'(x) to have roots as this would involve T Rolle and be more interesting
We love the Calvin and Hobbes shirt!
Courtesy of my mom, it’s my new favourite!