Question

The function, $f\left(x\right)={\int }_{-2}^{x}t(e^t-1)(t-1)(t-2{)}^3(t-3{)}^{5}dt$ has a local minima at

• A. $x=0$
• B. $x=2$
• C. $x=3$
• D. $x=5$

Answer 1

The correct option is C $x=3$

$f^{\prime }\left(x\right)=x(e^x-1)(x-1)(x-2{)}^3(x-3{)}^5$
$f^{\prime }\left(x\right)=0$ at $x=\{0,1,2,3\}$
$f^{\prime }\left(x\right)<0$, if $x<1$ or $2<x<3$
$f^{\prime }\left(x\right)>0$, if $x>3$ or $1<x<2$
$f\left(x\right)$ is minimum at $x=\{1,3\}$