Intervals of Increasing or Decreasing Step 2 of 2: The derivative of a function has already been found and factored to be $f^{\prime}(x)=(x-4)(x+6)$. Use the critical'values obtained in Step 1 to create intervals on a number line to test. Determine which intervals the original function (which we are not given and don't need) is increasing and which intervals it is decreasing, if any.
Solution
Step 1 ：The derivative of the function has been found and factored to be \(f^{\prime}(x)=(x-4)(x+6)\).
Step 2 ：The critical values of the function are the roots of the derivative, which are x = 4 and x = -6.
Step 3 ：These values divide the number line into three intervals: \((-\infty, -6)\), \((-6, 4)\), and \((4, \infty)\).
Step 4 ：A test point is picked in each interval and substituted into the derivative to determine whether the function is increasing or decreasing in that interval.
Step 5 ：If the derivative is positive, the function is increasing; if it is negative, the function is decreasing.
Step 6 ：The function is found to be increasing on the intervals \((-\infty, -6)\) and \((4, \infty)\), and decreasing on the interval \((-6, 4)\).
Step 7 ：\(\boxed{\text{The function is increasing on the intervals } (-\infty, -6) \text{ and } (4, \infty), \text{ and decreasing on the interval } (-6, 4).}\)
