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 ∫ _-1^1P(x)dx=18, then…

Let ${P}^{'}(x)=a(x-1)(x+1)=a({x}^{2}-1)$P′(x)=a(x−1)(x+1)=a(x2−1)

$$P(x)=a\int ({x}^{2}-1)dx+c$$P(x)=a∫(x2−1)dx+c

$$=a(\frac{{x}^{3}}{3}-x)+c$$=a(3x3​−x)+c

Now $P(-3)=0$P(−3)=0

$$\Rightarrow a(-\frac{27}{3}+3)+c=0$$⇒a(−327​+3)+c=0

$$\Rightarrow -6a+c=0\ldots (1)$$⇒−6a+c=0…(1)

Now $\displaystyle {\int }_{-1}^{1}(a(\frac{{x}^{3}}{3}-x)+c)dx=18$∫−11​(a(3x3​−x)+c)dx=18

$$=2c=18\Rightarrow c=9...(2)$$=2c=18⇒c=9...(2)

From (1)&(2) $\Rightarrow -6a+9=0\Rightarrow a=\frac{3}{2}$⇒−6a+9=0⇒a=23​

$$\Rightarrow P(x)=\frac{3}{2}(\frac{{x}^{3}}{3}-x)+9$$⇒P(x)=23​(3x3​−x)+9

Then, the Sum of coefficient$=\frac{1}{2}-\frac{3}{2}+9$=21​−23​+9

$$=8$$=8