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Math Help - Calculas: finding unknowns of polynominals

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    Calculas: finding unknowns of polynominals

    A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

    Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

    A) Find a, and b.

    B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?
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    Hello,

    Quote Originally Posted by Hibijibi View Post
    A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

    Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

    A) Find a, and b.

    B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?
    A local extremum (minimum or maximum) at a point of absciss m, is defined as following :
    f'(m)=0

    Here, m=-1 and f'(x)=\dots

    An inflection point is defined at a point of abscuss n, is defined as following :
    f''(n)=0

    Here, n=-2 and f''(x)=\dots
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    Quote Originally Posted by Hibijibi View Post
    A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

    Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

    A) Find a, and b.

    B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?
    Use f'(-1) = 0 and f''(-2) = 0

    f''(-2) = 24(-2) + 2a = 0 \Rightarrow a = -24

    f'(-1) = 12(-1)^2 + 2a(-1) + b = 0 \Rightarrow b = 2a - 12 = -60


    b) \int_{0}^{1} (4x^3 - 24x^2 - 60x + k) \, dx = 32

    (x^4 - 8x^3 - 30x^2 + kx)\bigg{|}_0^1  = 32

    (1 - 8 - 30 + k) = 32

    - 37 + k = 32

    k = 69

    Hopefully I have done no computational mistakes
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