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.

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- Apr 27th 2008, 06:24 AMlacy1104Rolle'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.

- Apr 27th 2008, 06:34 AMTheEmptySet

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

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

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

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

taking the derivative we get

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

setting equal to zero and solveing gives c=1 or

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