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Thread: 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

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

    $\displaystyle 0 = 3x^2 + 12x - 15$

    $\displaystyle 0 = x^2 + 4x - 5$

    $\displaystyle 0 = (x + 5)(x - 1)$

    $\displaystyle x = -5$ or $\displaystyle x = 1$.

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


    Since your minimum and maximum have the same absolute value,

    $\displaystyle |f(-5)| = |f(1)|$

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

    $\displaystyle |-125 + 150 - 75 + k| = |1 + 6 - 15 + k|$

    $\displaystyle |k - 50| = |k - 8|$

    $\displaystyle (k - 50)^2 = (k - 8)^2$

    $\displaystyle k^2 - 100k + 250 = k^2 - 16k + 64$

    $\displaystyle 186 = 84k$

    $\displaystyle k = \frac{186}{84}$

    $\displaystyle k = \frac{31}{14}$.
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