Question

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

Answer 1

The given function y is defined as,

y= x 2 ⋅ e −x

The derivative of the function y is given as,

y ′ = dy dx = d( x 2 ⋅ e −x ) dx =2x e −x − x 2 e −x =x e −x ( 2−x )

We have to find disjoint points in the real axis, so

dy dx =0 x e −x ( 2−x )=0

The solutions to the above equation are,

x=0 x=2

Hence, points x=0 and 2 divide the real axis into three intervals ( −∞,0 ), ( 0,2 ) and ( 2,∞ ).

For the intervals ( −∞,0 ) and ( 2,∞ ), y ′ is less than 0 as e −x is always positive.

Therefore, y is decreasing in intervals ( −∞,0 ) and ( 2,∞ ).

For the interval ( 0,2 ),

y ′ >0

As y ′ >0; therefore, y is strictly increasing in the interval ( 0,2 ).

Hence, the correct option is D.