Question

The set of all the critical points of $y=\underset{0}{\overset{x^2}{\int }}\frac{t^2-5t+4}{2+e^t}dt$ are

• A. $\{\pm 1,0\}$
• B. $\{\pm 2,0\}$
• C. $\{\pm 2,\pm 1,0\}$
• D. $\{3,2,1,0\}$

Answer 1

The correct option is C $\{\pm 2,\pm 1,0\}$

Let $y=f\left(x\right)=\underset{0}{\overset{x^2}{\int }}\frac{t^2-5t+4}{2+e^t}dt$
$\Rightarrow f^{\prime }\left(x\right)=\frac{x^4-5x^2+4}{2+e^{x^2}}2x-0=\frac{(x-1)(x+1)(x-2)(x+2)2x}{2+e^{x^2}}$
It is clear that $f^{\prime }\left(x\right)=0$ at $x=\pm 2,\pm 1,0$