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Math Help - Help on a challeging ABSOLUTE EXTREMA problem.

  1. #1
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    Help on a challeging ABSOLUTE EXTREMA problem.

    I find this problem really though. I did solve for it's derivative and then by EVT tried to find its crititcal points. I get (-π/2) and (π/2) for f'(x)=0.
    The problm is that I'm not sure if there are more solutions for f'(x).

    Please see function in problem 1 of attachment:


    Any help will be greatly apreciated.
    Thank you.
    Attached Thumbnails Attached Thumbnails Help on a challeging ABSOLUTE EXTREMA problem.-fotos.jpg  
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  2. #2
    MHF Contributor Amer's Avatar
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    Ok

    you mean the first question right

    the derivative of f(x) = 4x^3 + 12pie x cosx - 12 pie sinx + 3 pie^2 x

    df/dx= 12 x^2 + 12pie cosx - 12 pie x sinx -12 pie cosx + 3 pie ^2

    df/dx= 12x^2 - 12 pie x sinx + 3 pie^2

    df/dx= 3 ( 4x^2 - 4pie x sinx + pie^2 )

    ( -b +- (b^2 - 4ac)^1/2 )/2

    1/2(4 pie sinx +- (16pie^2 (sinx)^2 - 16 pie^2 )1/2 )

    1/2(4 pie sinx +- (16pie^2 ((sinx)^2 - 1) )^1/2 )

    1/2(4 pie sinx +- 4 pie( (sinx)^2 - 1 )1/2 )

    1/2(4 pie sinx +- 4 pie( -(cosx)^2 )1/2 )

    and -(cosx)^2 is a negative value for all values of x so you just have the value of x which make cosx zero like pie/2 + n pie n a natural number
    the solution is pie /2 and -pie/2
    there is no other solution

    my answer is very long
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  3. #3
    Senior Member Spec's Avatar
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    You'll eventually end up with the equation \sin^2 x = 1 which has the solution x=\frac{\pi}{2}+n\pi,\ n \in Z
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  4. #4
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    So how should I answer the question? since I'm not given any interval and there are infinately many solutions.
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  5. #5
    Senior Member Spec's Avatar
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    The answer I gave covers all the solutions.
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