Question

Find the absolute maximum and minimum values of each function over the indicated intervals.

a. f(x) = (4x) / (x²+1) over [-3,3]
b. g(x) = ln(x²+4) over [-3,1]

Answer 1

To find absolute maxima and minima, calculate the function's derivative, find critical points, evaluate the function at the endpoints and critical points, and compare values to determine the extrema over the given intervals.

Step-by-step explanation:

To find the absolute maximum and minimum values of functions over given intervals, you first calculate the derivative of the function and determine where the derivative is zero or undefined. These points, along with the endpoints of the interval, are potential candidates for local maxima and minima. Then, you evaluate the function at these points to find the absolute maximum and minimum values.

Part a: f(x) = (4x) / (x²+1) over [-3,3]

First, we calculate the derivative f'(x) and set it to zero to find critical points. Evaluate f(x) at these critical points as well as at the endpoints of the interval, x = -3 and x = 3. Compare these values to determine the absolute maximum and minimum.

Part b: g(x) = ln(x²+4) over [-3,1]

Again, we find the derivative g'(x), find critical points by setting the derivative to zero, and check for any discontinuities or undefined values within the interval. We then evaluate g(x) at the critical points and at the endpoints x = -3 and x = 1 to find the maxima and minima.