Let S=(-1,) and f:S R be defined as f(x)= _-1^x(e^t-1)^11(2t-1)^5(t-2)^7(t-3)^12(2t-10)^61…

Question

Let $S=(-1,\infty)$ and $f:S	o R$ 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: 27

Accepted Answer

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

Local minima at $x=\frac{1}{2},x=5$

Local maxima at $x=0,x=2$

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

Then ${p}^{2}+2q=16+11=27$

Solutions · 1