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...