To find: Absolute maximum or minimum.
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.