Let S = (−1,∞) and f: S → R be defined as

Question

Let \(\mathrm{S}=(-1, \infty)\) and \(\mathrm{f}: \mathrm{S} \rightarrow \mathbb{R}\) be defined as \(f(x)=\int\limits_{-1}^{x}\left(e^{t}-1\right)^{11}(2 t-1)^{5}(t-2)^{7}(t-3)^{12}(2 t-10)^{61} d t\)

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 \(p^{2}+2 q\) is ____.

Answer 1

Correct answer: 27

\(f(x)=\int\limits_{-1}^x\left(\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61}\right) d t\)

Using Lebnitz

\(f^{\prime}(x)=\left(e^x-1\right)^{11}(2 x-1)^5(x-2)^7(x-3)^{12}(2 x-10)^{61}\)

Point of maxima at \(\mathrm{x}=0,2\)

Point of minima at \(\mathrm{x}=\frac{1}{2}, 5\)

\(\mathrm{p}=0^2+2^2=4 \)

\(\mathrm{q}=\frac{1}{2}+5=\frac{11}{2} \)

\(\mathrm{p}^2+2 \mathrm{q}=16+11=27\)