Final answer:
The absolute maximum of the function g(x) = e^(-x⁴) on the interval -4 <= x <= 3 is 1, which occurs at x = 0.
Step-by-step explanation:
To find the absolute maximum of the function g(x) = e^(-x⁴) on the interval -4 <= x <= 3, we first need to find the critical points. These occur when the derivative of the function is equal to zero or does not exist. Taking the derivative of g(x) with respect to x gives us g'(x) = -4x³e^(-x⁴). Setting g'(x) equal to zero, we get -4x³e^(-x⁴) = 0. Since the exponential term e^(-x⁴) is always positive, we can see that the only critical point is x = 0. To determine if this critical point is a maximum or minimum, we can evaluate the function at x = 0 and the endpoints of the interval. Plugging these values into g(x), we find that g(-4) = e^(-256), g(0) = 1, and g(3) = e^(-81). Comparing these values, we can see that the absolute maximum of g(x) occurs at x = 0, with a value of g(0) = 1.