I am given two points on a cubic curve: a local maximum at $(-2,3)$ and a local minimum at $(1,0)$. I am to find an $f(x)=ax^3+bx^2+cx+d$.
So far, I've been able to ascertain only a little:
$f'(x) = A(x+2)(x-1) = 3ax^2 + 2bx +c$
therefore
$$
\begin{eqnarray}
2b &=& 3a \\
c &=& -4b = -6a \\
\end{eqnarray}
$$
However, I am unsure how to proceed. The textbook chapter did not cover this; it described ways to use $f(x)$ and it's derivatives to analyze the appearance of a graph, but nothing about finding an equation from part of it's derivative.
Best Answer
You have four linear equations and four unknown quantities: $$f(-2)=3,\ \ f(1)=0,\ \ f'(-2)=0, \mbox{ and } f'(1)=0$$ Thus
$$f(-2)=-8a+4b-2c+d=3$$$$f(1)=a+b+c+d=0$$$$f'(-2)= 12a-4b+c=0$$$$f'(1)=3a+2b+c=0$$