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Thread: find constants

  1. #1
    Senior Member euclid2's Avatar
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    find constants

    find constants a,b, and c such that the graph of $\displaystyle f(x)=x^3+ax^2+bx+c$ will increase to the point $\displaystyle (-3,18)$, decrease to the point (1,-14) and then continue increasing
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  2. #2
    Super Member Deadstar's Avatar
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    Quote Originally Posted by euclid2 View Post
    find constants a,b, and c such that the graph of $\displaystyle f(x)=x^3+ax^2+bx+c$ will increase to the point $\displaystyle (-3,18)$, decrease to the point (1,-14) and then continue increasing
    Find the derivative.

    $\displaystyle f'(x) = 3x^2 + 2ax + b$

    Set this equal to zero at the points $\displaystyle x=-3$ and $\displaystyle x=1$ which will be our critical points.

    We get $\displaystyle 27 - 6a + b = 0$ and $\displaystyle 3 + 2a + b = 0$

    Solving these simultaneous equations gives us...

    $\displaystyle a = 3$ and $\displaystyle b = -9$.

    You can check the second derivative to see that the points $\displaystyle (-3,18)$ and $\displaystyle (1,-14)$ are indeed a max and a min.

    Hence we have $\displaystyle f(x) = x^3 + 3x^2 - 9x + c$.

    Now just find a value of $\displaystyle c$ that solves...

    $\displaystyle -14 = 1 + 3 - 9 + c$ (for the point $\displaystyle (1,-14)$...)

    Graph of said function below after you find the $\displaystyle c$ value...
    Attached Thumbnails Attached Thumbnails find constants-untitled.jpg  
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