JEE PYQ: Application Of Derivatives Question 9
Question 9 - 2021 (18 Mar Shift 2)
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 $\int_{-1}^{1} P(x),dx = 18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to ______.
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Answer: 8
Solution
Let $P’(x) = a(x-1)(x+1) = a(x^2-1)$. Then $P(x) = a\left(\frac{x^3}{3} - x\right) + c$. Since $P(-3) = 0$: $-6a + c = 0$. Also $\int_{-1}^1 P(x),dx = 2c = 18 \Rightarrow c = 9 \Rightarrow a = \frac{3}{2}$. Sum of coefficients $= P(1) = \frac{3}{2}(\frac{1}{3} - 1) + 9 = -1 + 9 = 8$.
BETA
