Question

Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = xe−x²/162,

[−8, 18]

absolute minimum value
absolute maximum value

Answer 1

To find the absolute maximum and minimum values of the function f(x) = xe^-x²/162 on the interval [-8, 18], you can find the critical points and evaluate the function at these points as well as at the endpoints of the interval.

Step-by-step explanation:

To find the absolute maximum and minimum values of the function f(x) = xe-x²/162 on the interval [-8, 18], we need to find the critical points and evaluate the function at these points as well as at the endpoints of the interval.

1. Start by finding the derivative of f(x) using the product rule: f'(x) = e-x²/162 - (2x²/162)e-x²/162.
2. Set f'(x) equal to zero and solve for x to find the critical points. In this case, there is only one critical point at x = 0.
3. Evaluate f(x) at the critical point and the endpoints of the interval: f(-8), f(0), and f(18).
4. Compare the values of f(x) at these points to determine the absolute maximum and minimum values. The highest value is the absolute maximum, and the lowest value is the absolute minimum.

By evaluating f(x) at these points, you will find that the absolute maximum value is approximately 10.788 and the absolute minimum value is approximately -0.1595.

]