Question

Find the absolute maximum and absolute minimum values of the function, if they exist, on the indicated interval. 6) f(x) = x 4 - 32x 2 + 2; [-5, 5]

Answer 1

we have

f'(x) = 0 for the critical point,

the behavior of f'(x),

	x=-5	-5<x<-4	x=-4	-4<x<0	x=0	0<x<4	x=4	4<x<5	x=5
sign	NA	-	0	+	0	-	0	+	NA
behavior	maximum	decreasing	minimum	increasing	maximum	decreasing	minimum	increasing	maximum

the minimum at x = -4 is,

the minimum at x = 4 is,

the maximum at x = -5 is,

the maximum at x = 5 is,

the maximum at x = -5 is,

hence

the absolute minimum at x = -4 is -173 and at x = 4 is -173,

the absolute maximum at x = 0 is 2.