Calculus

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A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx + k?
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 the values of a and b

b) If ∫ (from 0 to 1) f(x) dx = 32, what is the value of k?

I found a=24 and b=36
i don't know about B)

  • Calculus -

    From a=24,b=36,
    we find
    f'(x)=0 has roots at x=-3 and -1, so that checks.
    f"(x)=0 has a root at x=-2, so that checks too.

    To solve B, we integrate
    f(x):=4*x^3+24*x^2+36*x+k
    from 0 to 1:
    I=∫f(x)dx
    =x^4+8*x^3+18*x^2+k*x
    Evaluate I from 0 to 1 to give k+27
    But since is given as I=32, so
    I=32=k+27
    Solve for k.

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