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Math Help - Critical Numbers/Extrema of a function

  1. #1
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    Critical Numbers/Extrema of a function

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  2. #2
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    Hello, zaurm!

    The problem is straight-forward.
    Exactly where is your difficulty?


    a) Find the critical numbers of: . f(x) \:=\:\frac{1}{2}\,x - \sin x .in the interval [0,\pi]
    Equate the derivative to zero and solve . . .

    . . f'(x) \:=\:\frac{1}{2}-\cos x \:=\:0 \quad\Rightarrow\quad \cos x \:=\:\tfrac{1}{2}\quad\Rightarrow\quad x \:=\:\frac{\pi}{3}




    b) Find the exrema of f(x) .in the inverval [0,\pi]
    . . \begin{array}{ccccccccc}f(0) &=& 0 - \sin 0 &=& 0 \\<br />
f(\frac{\pi}{3}) &=& \frac{1}{2}\!\cdot\!\frac{\pi}{3} - \sin\frac{\pi}{3} &=& \frac{\pi}{6} - \frac{\sqrt{3}}{2} & \leftarrow \text{ minimum}\\<br />
f(\pi) &=& \pi -\sin\pi &=& \pi & \leftarrow \text{ maximum}<br />
\end{array}

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