Thread: prove there is a c_1

Suppose f: [0,2]-> R is differentiable. f(0)=0, f(1)=2 and f(2)=2. Prove that

1. There is such that f'( ) =0
2. There is such that f'(c_2)=2 and
3. There is such that f'( =1/2

For (1) and (2), use the mean value theorem on the intervals [1,2] and [0,1] respectively. Since a derivative always has the intermediate value property (Darboux's theorem), it follows that there is a between and where the derivative is 1/2.

Originally Posted by maddas

For (1) and (2), use the mean value theorem on the intervals [1,2] and [0,1] respectively. Since a derivative always has the intermediate value property (Darboux's theorem), it follows that there is a between and where the derivative is 1/2.

Hmm..that makes sense lol thanks !