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Math Help - A cubic Polynomial

  1. #1
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    A cubic Polynomial

    Hi,
    There is a problem:
    For f(x)=(x^3)+(6x^2)-15x+k
    The absolute minimum and absolute maximum values on the interval [-10,2] have the same absolute value. Find the value of k.
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  2. #2
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    Quote Originally Posted by spred View Post
    Hi,
    There is a problem:
    For f(x)=(x^3)+(6x^2)-15x+k
    The absolute minimum and absolute maximum values on the interval [-10,2] have the same absolute value. Find the value of k.
    Absolute minimums and maximums occur where the derivative is 0.

    So

    f'(x) = 3x^2 + 12x - 15

    0 = 3x^2 + 12x - 15

    0 = x^2 + 4x - 5

    0 = (x + 5)(x - 1)

    x = -5 or x = 1.

    This means the critical values are f(-5) and f(1).


    Since your minimum and maximum have the same absolute value,

    |f(-5)| = |f(1)|

    |(-5)^3 + 6(-5)^2 -15(-5) + k| = |1^3 + 6(1)^2 - 15(1) + k|

    |-125 + 150 - 75 + k| = |1 + 6 - 15 + k|

    |k - 50| = |k - 8|

    (k - 50)^2 = (k - 8)^2

    k^2 - 100k + 250 = k^2 - 16k + 64

    186 = 84k

    k = \frac{186}{84}

    k = \frac{31}{14}.
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