Question

In Exercises \(31-34,\) find the extreme values of the function on the interval and where they occur. $$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$

Short answer

The extreme values of the function \(g(x)=|x-1|-|x-5|\) on the interval -2 ≤ x ≤ 7 are to be evaluated at the critical points and interval boundaries. Then, identify which are the maximum and minimum values and where they occur.

Step-by-step solution

Determine the critical points

To start, find the critical points. These will be found where \(x - 1 = 0\) or \(x - 5 = 0\), since that's where the function changes its direction. Solving these equations will give \(x = 1\) and \(x = 5\). Lying in the interval -2 ≤ x ≤ 7, these are valid critical points.

Evaluate the function at critical points and interval boundaries

Evaluate the function at the critical points x = 1, x = 5 and at the boundaries of the interval x = -2, x = 7. Remember that the absolute value of a number is its distance from zero, so whether it's negative or positive, distance is always positive.

Find the extreme values

Having evaluated the function at these important points, now identify the highest and lowest values obtained, these will be the maximum and minimum, or extreme values of the function on the given interval. In order to decide where the extreme values occur, look at the 'x' values that gave you these extreme 'y' values.