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Math Help - finding the max and min values

  1. #1
    Junior Member
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    finding the max and min values

    Let a be a positive constant and define f by f(x)=(a^2)(x^2)-(x^4). find the maximum and minimum values of f on [0,2a].

    f(x)=(a^2)(x^2)-(x^4)
    (f^1)(x) = (a^2)(2x)-4x^3
    = 2(a^2)x - 4x^3
    = 2x((a^2)-2x^2)

    2x=0 (a^2)-2x^2 =0
    x=0 2x^2 = a^2
    x^2=(a^2)/2
    x= a/sqrt2
    The critical points are 0, 2a, and a/sqrt2
    To find the min and the max values we would substitute the critical points in f(x). Is this right so far?
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  2. #2
    MHF Contributor

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    Yes, it is. Since the problem is asking for the maximum and minimum values, the "global" max and min, you need only evaluate the function at those three points to find which gives the largest value of x and which the smallest.
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