Reconstructing the parameters of a cubic

Text

Find the coefficients $a,b,c$ of $y=ax^{3}+bx^{2}+cx$ given that it has a relative maximum at $x=-1$ with $y=2$ and a relative extremum at $x=1$.

Solution

$y'=3ax^{2}+2bx+c$. Conditions $y'(-1)=0$ and $y'(1)=0$: $3a-2b+c=0$ and $3a+2b+c=0$. Subtracting: $4b=0\Rightarrow b=0$, so $c=-3a$. Condition $y(-1)=2$: $y(-1)=-a-c=2a=2\Rightarrow a=1$, $c=-3$. The function is $y=x^{3}-3x$. $\boxed{a=1,\ b=0,\ c=-3}$