Question

Find the maximum and minimum value of the function $f\left(x\right)=x^2{e}^x$.

Answer 1

$f\left(x\right)=x^2{e}^x$

$e^x>0\forall x\in R$

also $x^2>0\forall x\in R$

$\therefore$ minimum value ocues at $x=0$

$\therefore f\left(0\right)=(0{)}^L{e}^0=0$

$\therefore$ minimum value of $f\left(x\right)=0$

$f^{\prime }\left(x\right)=2x{e}^x+x^2{e}^x$

$f^{\prime }\left(x\right)=2e^x+2x{e}^x+2x{e}^x+x^2{e}^x$

$f\left(x\right)$ is on increasing function.

So, $f\left(x\right)$ reads infinity as $x$axis.

$\therefore f\left(x\right)\in [0,\infty ]$