Question

The interval in which y=x^2e^-x is increasing is (A)(-∞,∞) (B) (−2, 0) (C)(2,∞) (D) (0, 2)

Answer 1

See your answer in attachment!!

Answer 2

given, $\bf{y=x^2e^{-x}}$
differentiate y with respect to x,
$\bf{\frac{dy}{dx}=\frac{d}{dx}(x^2e^{-x})}\\\\=\bf{x^2\frac{de^{-x}}{dx}+e^{-x}\frac{dx^2}{dx}}\\\\=\bf{-x^2e^{-x}+e^{-x}2x}\\=\bf{xe^{-x}(2-x)}$

for increasing, dy/dx > 0
e.g., $xe^{-x}(2-x)$ > 0
as you know, e^-x is an exponential function, we know exponential function is always positive for all real numbers .
so, remove it from inequality, then x(2-x) > 0
e.g., 0 < x < 2

therefore, f is increasing in (0, 2)
hence, option (D) is correct.