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

Correct Answer: 27

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:

\[ p^2 + 2q = 16 + 11 = 27 \]