absolute maximum

Question

Consider the function f(x) = x^3e^6x -2< x< 2

The absolute maximum value is ... and this occurs at x equals .....

The absolute minimum value is .... and this occurs at x equals .

Answer 1

I assume that you mean -2 ≤ x ≤ 2?

f(x) = x³e^6x ( f is continuous on [-2,2] so the absolute extrema occur at a critical point or endpoint)

f'(x) = 3x²e^6x + 6x³e^6x = 3x²e^6x(1 + 2x) = 0 when x = 0 or -1/2

f(0) = 0

f(2) = 8e¹² (abs max)

f(-2) = 4e^-12

f(-1/2) = -(1/8)e^-3 (abs min)