Question

The interval in which $y=x^2{e}^{-x}$ is increasing with respect to x is

a) $(-\infty ,\infty )$
b) (-2, 0)
c) $(2,\infty )$
d) (0.2)

Answer 1

Given, $y=x^2{e}^{-x}\Rightarrow \frac{dy}{dx}=x^2{e}^{-x}(-1)+e^{-x}\left(2x\right)=x{e}^{-x}(-x+2)=x(2-x)e^{-x}$
For increasing function, $\frac{dy}{dx}>0\Rightarrow x{e}^{-x}(2-x)>0$
$\Rightarrow x>0$ or $2-x>0\Rightarrow x>0$ or $-x>-2\Rightarrow x>0$ or $x<-2$
Because e^{-x} cannot be zero, clearly it is increasing in (0,2) So, correct answer is (d).