Final answer:
To find the absolute maximum and minimum of the function f(x) = xe^{-x^2/72} on the interval [-3, 12], calculate its derivative, identify critical points and the endpoints of the interval.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = xe−x2/72 on the interval [−3, 12], we first need to identify any critical points where the derivative of f(x) is zero or undefined and then evaluate the function at the critical points and at the endpoints of the interval.
The derivative of the function f(x) is found using the product rule and chain rule of differentiation. After finding the derivative, set it equal to zero to solve for x. These x-values are your possible critical points. Evaluate the function f(x) at the critical points and at the endpoints of the interval.
The largest value will be the absolute maximum, and the smallest value will be the absolute minimum. Find the derivative of f(x). Solve for x when the derivative is zero (critical points). Evaluate f(x) at critical points and interval endpoints [−3, 12]. Compare the values to determine the absolute maximum and minimum.