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In an effort to not waste the time you generously provide to help others, my question is:

A cubic polynomial function is defined by f(x)=4x^(3)+ax^(2)+bx+k, where a, b, and k are constants. The function has a local minimum at x=-1, and the graph of f has a point of inflection at x=-2. What are the values of a and b?

Since f(x) = 4x^3 + ax^2 + bx + k, f'(x) = 12x^2 + 2ax + b and f"(x) = 24x + 2a.

Having an inflection point at x=-2 says that f"(-2) = 0.

This means 24(-2) + 2a = 0, and that means -48 = -2a, so a = 24.

This means f'(x) = 12x^2 - 48x + b. To have a minimum at x=-1 we would need

12 + 48 + b = 0. This would say that 60 + b = 0, so b = -60.

This says that the equation is then f'(x) = 12x^2 + 24x - 60 = 12(x^2 + 2x - 5).

There is no way we can find k, and it is not asked for, so that's it.

Calculus

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