Intervals of Change

Increasing Intervals are the ranges of values in which the function INCREASES with respect to the increase in \(x\). Conversely, Decreasing Intervals are the ranges of values in which the function DECREASES with respect to the increase in \(x\).

In order to determine whether a function is increasing or decreasing, we can first find its derivative. We can then conduct the First Derivative Test. This involves solving \(f'(x) = 0\) or \(f'(x) = \text{Undefined}\) by finding test \(x\)-values which can indicate the intervals the function is increasing or decreasing. The test value can be picked from any value within an interval between critical points or end points.

The First Derivative Test states:

  • If \(f'(x) > 0\), \(f(x)\) increases (or has a positive slope)
  • If \(f'(x) < 0\) \(f(x)\) decreases (or has a negative slope)

We can then identify the Max or Min points if there is a sign change between intervals. If it changes from positive to negative, then that point is a Max. If it changes from negative to positive, then that point is a Min. If there is no sign change between intervals, then that point is neither.


Example

For the function \(f(x) = 2x^3 + 3x^2 -36x +5\):

  1. Find the intervals of increase and decrease.
  2. Graph \(f(x)\) and \(f'(x)\) and explain how they indicate intervals of increase and decrease.

i. First, we can differentitate the function:

\(f(x) = 2x^3 + 3x^2 -36x +5\)

\(f'(x) = (3)(2x^2) + (2)(3x) -(1)(36) + (0)(5)\)

\(f'(x) = 6x^2 + 6x -36\)

Next, we can factor the derivative to solve for its critical numbers:

\(f'(x) = 6(x^2 + x -6)\)

\(f'(x) = 6(x + 3)(x-2)\)

\(x_1 = -3\)

\(x_2 = 2\)

After, we can conduct the First Derivative Test to determine the increasing and decreasing intervals:

Interval Test \(x\)-value \(f'(x)\) Conclusion
\(\textcolor{green}{(-∞,-3)}\) \(-4\) \(f'(-4)=36\) \(f\) is Increasing
\(\textcolor{red}{(-3,2)}\) \(0\) \(f'(0)=-36\) \(f\) is Decreasing
\(\textcolor{green}{(2,∞)}\) \(3\) \(f'(3) = 36\) \(f\) is Increasing

Therefore, we can determine that the Intervals of Increase are:

\(\textcolor{green}{x∈(-∞,-3)}\)

\(\text{AND}\)

\(\textcolor{green}{x∈(2,∞)}\)

We can also determine that the Interval of Decrease is:

\(\textcolor{red}{x∈(-3,2)}\)

ii. We can graph \(f(x)\) as such:

Graph of cubic function with green portions representing intervals of increase and red portions represeting intervals of decrease.

From this graph, we can determine:

  • \(f(x)\) moving upward with respect to \(x\) means that \(f(x)\) is increasing
  • \(f(x)\) moving downward with respect to \(x\) means that \(f(x)\) is decreasing

We can graph \(f'(x)\) as such:

Graph of quadratic function with green and red portions respectively representing components above and below x-axis.

From this graph, we can identify:

  • \(f'(x)\) above the \(x\)-axis means that \(f(x)\) is increasing
  • \(f'(x)\) below the \(x\)-axis means that \(f(x)\) is decreasing

Find the intervals of increase and decrease for the function \(f(x) = \cfrac{6-2x}{x^2-4}\).

Try these questions: