# Math Help - Rolle's theorem

1. ## Rolle's theorem

16. Define by . Show that f satisfies the conditions of Rolle's theorem and find c such that f'(c) = 0.

2. Originally Posted by lacy1104
16. Define by . Show that f satisfies the conditions of Rolle's theorem and find c such that f'(c) = 0.

Well, what are the hypothsis of Rolle's Theorem.

f is cont on [0,2] and differentiable on (0,2).

Lastly $f(0)=\sqrt{2(0)-0^2}=0,f(2)=\sqrt{2(2)-(2)^2}=0$

So there exits a $c \in (0,2)$ such that $f'(c)=0$

taking the derivative we get

$\frac{df}{dx}=\frac{1}{2}\cdot \frac{2-2x}{\sqrt{2x-x^2}}=\frac{1-x}{\sqrt{2x-x^2}}$

setting equal to zero and solveing gives c=1 or

$f'(c)=0 \mbox{ when } c=1$