Question

Let $S = (-1, \infty)$ and $f : S \rightarrow \mathbb{R}$ be defined as $f(x) = \int_{-1}^{x} (e^t - 1)^{11} (2t - 1)^5 (t - 2)^7 (t - 3)^{12} (2t - 10)^{61} \, dt$ 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 $p^2 + 2q$ is ______

Answer 1

Consider the derivative:

$f'(x) = (e^{x-1})^{11} (2x - 1)^9 (x - 2)^7 (x - 3)^{12} (2x - 10)^{61}$

Analyzing the sign changes, we observe local minima at:

$x = \frac{1}{2}, \, x = 5$

And local maxima at:

$x = 0, \, x = 2$

Calculating values:

$p = 0^2 + 2^2 = 4, \quad q = \frac{1}{2} + 5 = \frac{11}{2}$

Therefore:

$p^2 + 2q = 16 + 11 = 27$