1. Mean Value Problem

Left f(x) be a function that is continuous on the interval [-5,5] with continuous 1st and 2nd derivatives on (-5,5). f(-5)=3, f(0)=-2 and f(5)=8.

1. Show that there is a point a in (-5, 0) such that f'(a) <0

2. Show that there is a point b in (0, 5) such that f'(b) <0

3. Using the results from parts 1 & 2, must f have a critical number in (-5, 5)? 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. [QUOTE=ebonyscythe;83912]Left f(x) be a function that is continuous on the interval [-5,5] with continuous 1st and 2nd derivatives on (-5,5). f(-5)=3, f(0)=-2 and f(5)=8.

1. Show that there is a point a in (-5, 0) such that f'(a) <0
Intermediate value theorem on f' applied to -5 and 0 because f'(-5)=3 and f'(0)=-2.

2. Show that there is a point b in (0, 5) such that f'(b) <0
I think you mean f'(b)>0 it is the same idea.
3. Using the results from parts 1 & 2, must f have a critical number in (-5, 5)? Explain why f must have a critical number or give an example of a function that satisfies the criteria but has no critical number.
Intermediate value theorem again. This time on 'a' and 'b'.