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Math Help - Rolle's theorem

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    Rolle's theorem

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