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...