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Math Help - Absolute max and min

  1. #1
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    Absolute max and min

    Find the absolute max and min value(s) for :

    a) f(x) = -x^2 + 4x + 7

    f'(x) = -2x + 4

    x = 2
    y = 11

    critical point (2,11)

    No absolute min or max value


    b) f(x) = -x^{4} + 4x^{3}

    f'(x) = -4x^{3} + 12x^2

    x = 0
    y = 0

    x = 3
    y = 27

    critical points: (0,0) and (3,27)

    Absolute min at (0,0)
    Absolute max at (3,27)

    c)
    f(x) = x + x^{-1} , x > 0

    f'(x) = 1 - \frac{1}{x}
    0 = \frac {x - 1}{x}

    x = 1
    y = 2

    critical point (1,2)

    since  x > 0, there is no absolute max
    absolute min value is (1,2)
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Macleef View Post
    Find the absolute max and min value(s) for :

    a) f(x) = -x^2 + 4x + 7

    f'(x) = -2x + 4

    x = 2
    y = 11

    critical point (2,11)

    No absolute min or max value


    b) f(x) = -x^{4} + 4x^{3}

    f'(x) = -4x^{3} + 12x^2

    x = 0
    y = 0

    x = 3
    y = 27

    critical points: (0,0) and (3,27)

    Absolute min at (0,0)
    Absolute max at (3,27)

    c) f(x) = x + x^{-1} , x > 0

    f'(x) = 1 - \frac{1}{x}
    0 = \frac {x - 1}{x}

    x = 1
    y = 2

    critical point (1,2)

    since  x > 0, there is no absolute max
    absolute min value is (1,2)
    How is x=2 not a max?

    since f(x)=-x^2+4x+7\Rightarrow{f'(x)=-2x+4}\Rightarrow{f''(x)=-2}

    you found the critical point and since f''(x)<0,\forall{x}\in\mathbb{R} we have that (2,11) is a relative maximum
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  3. #3
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    the critical point for a) isn't the absolute max or min ... there's only one coordinate point so I just assumed it doesn't mean anything?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Macleef View Post
    the critical point for a) isn't the absolute max or min ... there's only one coordinate point so I just assumed it doesn't mean anything?
    Oh it means something alright...to find absolute extrema you find all the relative extrema...

    Now you test all the x valuse for relative extrema and all the x values of the endpoints of your interval and the value that when put in f(x) yields the larges result is the absolute maximum point and the lowest is obviously the absolute minimum point
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