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  • 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 , then the s

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)d(x) =18, then the sum of all coefficients of the polynomial P(x) is equal to ______
Option: 1 38
Option: 2 24
Option: 3 16
Option: 4 8

Answers (1)

best_answer

\\P^{\prime}(x)=a(x+1)(x-1) \\ \therefore P(x)=\frac{a x^{3}}{3}-a x+C \\ P(-3)=0(\text { given }) \\ \Rightarrow a(-9+3)+C=0 \\ \Rightarrow 6 a=C\;\;\;\;\;\;\;\;\;\;\;\ldots(1)

\\\text { Also, } \int_{-1}^{1} \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}=18 \\\Rightarrow \int_{-1}^{1}\left(\mathrm{a}\left(\frac{\mathrm{x}^{3}}{3}-\mathrm{x}\right)+\mathrm{C}\right) \mathrm{d} \mathrm{x}=18

=2 \mathrm{C}=18 \Rightarrow \mathrm{C}=9

from (1)

\\a=\frac{3}{2} \\ \therefore P(x)=\frac{x^{3}}{2}-\frac{3}{2} x+9 \\ \text { Sum of co-efficient }=-1+9=8

 

Posted by

himanshu.meshram

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