Question

Let $P(x)$ be a real polynomial of degree 3 which vanishes at $x=-3.$ Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and $\underset{-1}{\overset{1}{\int }}P(x)dx=18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to ________.

Answer 1

Answer: 8

Let $p^{\prime }(x)=a(x-1)(x+1)=a\left(x^2-1\right)$
$p(x)=a\int \left(x^2-1\right)dx+c$
$=a\left(\frac{x^3}{3}-x\right)+c$
Now $p(-3)=0$
$\Rightarrow a\left(-\frac{27}{3}+3\right)+c=0$
$\Rightarrow -6a+c=0\dots (1)$
Now $\underset{-1}{\overset{1}{\int }}\left(a\left(\frac{x^3}{3}-x\right)+c\right)dx\mid =18$
$=2c=18\Rightarrow c=9\dots$(2)
$\Rightarrow$ from $(1)\&(2)$
$\Rightarrow -6a+9=0$
$\Rightarrow a=\frac{3}{2}$
$\Rightarrow p(x)=\frac{3}{2}\left(\frac{x^3}{3}-x\right)+9$
sum of coefficient
$=\frac{1}{2}-\frac{3}{2}+9$
$=8$