Let $$S = [-1, \infty)$$ and $$f: S \to \mathbb{R}$$ 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 square 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 ________

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

Critical points and their sign changes ($$f'(x)$$ changes sign only at odd powers):

• $$x = 0$$ (odd power): sign changes $$+$$ to $$-$$ $$\implies$$ Local Maxima

• $$x = \frac{1}{2}$$ (odd power): sign changes $$-$$ to $$+$$ $$\implies$$ Local Minima

• $$x = 2$$ (odd power): sign changes $$+$$ to $$-$$ $$\implies$$ Local Maxima

• $$x = 3$$ (even power): no sign change

• $$x = 5$$ (odd power): sign changes $$-$$ to $$+$$ $$\implies$$ Local Minima

• Maxima points: $$p = 0^2 + 2^2 = 4$$

• Minima points: $$q = \frac{1}{2} + 5 = \frac{11}{2}$$

$$p^2 + 2q = 4^2 + 2\left(\frac{11}{2}\right) = 16 + 11 = 27$$