The Extreme Value Theorem guarantees that a continuous function, such as this one, on a closed interval has an absolute min and absolute max on that interval. It further specifies that those extrema will occur at a critical point for the function (f'(x) = 0 or f'(x) undefined) or at one of the endpoints.
For this question, we find f(-1), f(1), and y-values corresponding to any places on the interval where f'(x) = 0:
f(-1) = - 1/e4 , f(1) = e4
By product rule, f'(x) = 3x2e4x + 4x3e4x = 0
x2e4x(4x + 3) = 0 ; x = 0 , x = - 3/4
Plugging in will confirm f(-3/4) is the absolute min, while f(1) is the absolute max.