Suppose f:[a,b]->R satisfies the following properties:
1. f is continuous on [a,b] and continuously differentiable on (a,b)
2. f(a) and f(b) are positive numbers
3. f(x)=0 for some x in (a,b)
Prove that there exists some c in (a,b) such that f'(c)=0.
I understand what the problem is asking and can draw a descriptive picture, but I'm having trouble putting it in general terms. Since f(a) and f(b) are both positive numbers, and there is some x such that f(x)=0, then the slope from f(a) to f(x) must be negative, and the slope from f(x) to f(b) must be positive. Since the function is continuous and continuously differentiable, then the derivative function must also be continuous.
Can you show me how to prove that f'(c)=0?