# Rolle's theorem

• April 27th 2008, 06:24 AM
lacy1104
Rolle's theorem
16. Define http://answerboard.cramster.com/Answ...7096666538.gifby http://answerboard.cramster.com/Answ...0672545628.gif. Show that f satisfies the conditions of Rolle's theorem and find c such that f'(c) = 0.
• April 27th 2008, 06:34 AM
TheEmptySet
Quote:

Originally Posted by lacy1104
16. Define http://answerboard.cramster.com/Answ...7096666538.gifby http://answerboard.cramster.com/Answ...0672545628.gif. 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

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