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Thread: Maximum value

  1. #1
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    Maximum value

    find the maximum value of
    (cos(cos x))^2 +(sin(sin x))^2
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  2. #2
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    Quote Originally Posted by ayushdadhwal View Post
    find the maximum value of
    (cos(cos x))^2 +(sin(sin x))^2
    $\displaystyle u = \cos{x}$ , $\displaystyle v = \sin{x}$

    $\displaystyle y = \cos^2(u) + \sin^2(v)$

    $\displaystyle \dfrac{dy}{dx} = -2\cos(u)\sin(u) \cdot \dfrac{du}{dx} + 2\sin(v)\cos(v) \cdot \dfrac{dv}{dx}$

    take it from here?
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    Quote Originally Posted by skeeter View Post
    $\displaystyle u = \cos{x}$ , $\displaystyle v = \sin{x}$

    $\displaystyle y = \cos^2(u) + \sin^2(v)$

    $\displaystyle \dfrac{dy}{dx} = -2\cos(u)\sin(u) \cdot \dfrac{du}{dx} + 2\sin(v)\cos(v) \cdot \dfrac{dv}{dx}$

    take it from here?
    not getting
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  4. #4
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    Quote Originally Posted by ayushdadhwal View Post
    not getting
    Are you even trying? You were told that u= cos(x). What is du/dx? You were told that v= sin(x). What is dv/dx?
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ayushdadhwal View Post
    find the maximum value of (cos(cos x))^2 +(sin(sin x))^2

    The function

    $\displaystyle f(x)= (\;\cos (\cos x)\^2+(\;\sin (\sin x)\^2$

    is even and periodic with period $\displaystyle \pi$ so, we only need to find the maximum on $\displaystyle [0,\pi/2]$ .

    The values of $\displaystyle f$ at the endpoints of $\displaystyle [0,\pi/2]$ are:

    $\displaystyle f(0)=\cos^2 1<\sin^2 1=f(\pi/2)$

    Prove that there are no singular points in $\displaystyle (0,\pi/2)$ so, the absolute maximun of $\displaystyle f$ is $\displaystyle \sin^2 1$ and the absolute minimum $\displaystyle \cos^2 1$ .
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