Let and be defined as Let Sum of squares of the values of , where attains local maxima on and Sum of the values of where attains local minima on . Then, the value of is

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Solution

Q.
Let $S = (- 1, \infty)$ and $f : S \to R$ be defined as $f (x) = \int_{- 1}^{x} {(e^{t} - 1)}^{11} {(2 t - 1)}^{5} {(t - 2)}^{7} {(t - 3)}^{12} {(2 t - 10)}^{61} d t .$ Let $p =$ Sum of squares of the values of $x$ , where $f (x)$ attains local maxima on $S,$ and $q =$ Sum of the values of $x,$ where $f (x)$ attains local minima on $S$ . Then, the value of $p^{2} + 2 q$ is ____________ . [2024]

Ans.
(27)

Given,

$f (x) = \int_{- 1}^{x} {(e^{t} - 1)}^{11} {(2 t - 1)}^{5} {(t - 2)}^{7} {(t - 3)}^{12} {(2 t - 10)}^{61} d t$

$f' (x) = {(e^{x} - 1)}^{11} {(2 x - 1)}^{15} {(x - 2)}^{7} {(x - 3)}^{12} {(2 x - 10)}^{61}$

$So, x = 0, x = \frac{1}{2}, x = 2, x = 3, x = 5$

$∴ p = 0 + 4 = 4$

$q = \frac{1}{2} + 5 = \frac{11}{2}$

$∴ p^{2} + 2 q = 16 + 11 = 27$

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