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$[-1,1].$