1. ## Calculas Problem, difficult

Let f(x) be a function that is continuous on the interval [-10, 10], has continuous first and second derivatives on (-10, 10), and has values f(-10)=4, f(0)=-2, and f(10)=8.

a) Show that there is a point a in (-10, 10) such that f'(a)<0

b) Show that there is a point b in (0, 10) such that f'(b)>0

c) Using the results from parts a & b, must f have a critical number in (-10, 10? Explain why f must have a critical number or give an example of a function that satisfies the criteria but has no critical number.

2. Originally Posted by Hibijibi
Let f(x) be a function that is continuous on the interval [-10, 10], has continuous first and second derivatives on (-10, 10), and has values f(-10)=4, f(0)=-2, and f(10)=8.

a) Show that there is a point a in (-10, 10) such that f'(a)<0

b) Show that there is a point b in (0, 10) such that f'(b)>0

c) Using the results from parts a & b, must f have a critical number in (-10, 10? Explain why f must have a critical number or give an example of a function that satisfies the criteria but has no critical number.