Answer to Question #295389 in Calculus for DAN

Question

Given f(x)=x^7 -x^5 -x^4 +2x +1 on interval [-1,1] show that there's at least one critical point on this interval.

Answer 1

$f(x)=x^7 -x^5 -x^4 +2x +1$

The function $f(x)$ is continuous and differentiable on $\R$ as polynomial.

$f'(x)=7x^6-5x^4-4x^3+2$ $f'(0)=2>0$

$f'(0.8)=7(0.8)^6-5(0.8)^4-4(0.8)^3+2$

$=-0.260992<0$

Then by the Intermediate Value Theorem, there exists a number $c\in (0, 0.8)$ such that $f'(c)=0.$

Therefore there's at least one critical point on the interval$[-1,1].$