To determine

To find: Absolute maximum or minimum.

Answer

Answer: f-2=-2e

Explanation

Given:fx=xex2, on [-3,1].

Let us find out the first derivative of f and equate it to zero:

f'x=x2ex2+ex2=0 which implies ex21+x2=0, which implies x=-2. If x<-2, then x2<-1, i.e., x2+1<0, i.e., ex2x2+1<0(because exponential function is always positive), i.e., f'x<0 for x<-2.

Also, if x>-2, then x2>-1, i.e., x2+1>0, i.e., ex21+x2>0, i.e., f'x>0 for x>-2. By First Derivative Test, we get that x=-2 is the point of minima and f-2=-2e.

Conclusion: f-2=-2e