Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B), (D)

The correct options are
B $1$
D $3$

$f\left(x\right)={\int }_{-1}^{x}t(e^t-1)(t-1)(t-2{)}^3(t-3{)}^{5}dt$
$\Rightarrow f^{\prime }\left(x\right)=x(e^x-1)(x-1)(x-2{)}^3(x-3{)}^5$
The critical points are $0,1,2,3.$
Sign scheme of $f^{\prime }\left(x\right)$
Clearly $x=1$ and $x=3$ are the points of minima.