Question

The interval in which is increasing is (A) (B) (−2, 0) (C) (D) (0, 2)

Answer 1

The interval in which $\bf{y=x^2e^{-x}}$ is increasing is
A. (– ∞, ∞)
B. (– 2, 0)
C. (2, ∞)
D. (0, 2)

solution :- we know function f(x) is increasing on interval [a,b] only when f'(x) > 0 in interval [a,b].

$y=x^2e^{-x}\\\\\text{differentiate with respect to x}\\y'=2xe^{-x}+x^2(-e^{-x})\\y'=xe^{-x}(2-x)>0$

here $e^{-x}$ is an exponential function.
so, it is always positive .
e.g., x (positive)(2 - x ) > 0
=> x(2 - x) > 0
=> 0 < x < 2

hence, f(x) = $x^2e^{-x}$ is increasing on (0,2).

hence, option (d) is correct.