Question

Find the intervals in which f(x)=x²e⁻ˣ is increasing or decreasing. x ∈ R.

Answer 1

any function , f(x) is a increasing function in interval (a,b) only when f'(x) > 0 in (a,b) while decreasing only when f'(x) < 0 in (a ,b)

here, $f(x)=x^2e^{-x}$
differentiate with respect to x,
$f'(x)=2xe^{-x}-e^{-x}x^2$
now, f'(x) = 0
$2x.e^{-x}-e^{-x}x^2=0$
$xe^{-x}(2-x)=0$
x = 0 , 2

case 1 :- x < 0 , f'(x) < 0
so, f(x) is strictly decreasing in (-∞ , 0)

case 2 :- 0 < x < 2 , f'(x) > 0
so,f(x) is strictly increasing in (0, 2)

case 3 :- x > 2 , f'(x) < 0.
so, f(x) is strictly decreasing in (2, ∞)

finally, function , f(x) is strictly increasing in (0,2) while strictly decreasing in (-∞ , 0) U (2, ∞)

Answer 2

Dear Student:

F(x)=x²e⁻ˣ

For,maximum and minimum  

We will find df/dx and then equate to zero.

If df/dx>o then f is strictly increasing.

If df/dx<0 then strictly decreasing.

And then again proceed for second derivative and see it is negative or positive.

If it is positive then f will be minimum

And if negative then f maximum

See the attachment: