Intervals of Concavity

Definitions

Definition

A function is concave up on an interval if its derivative is increasing on the interval.

Definition

A function is concave down on an interval if its derivative is decreasing on the interval.

What the Sign of \(f''\) Tells Us About \(f\).

• If \(f''(x) > 0\) for all \(x\) on the interval \((a,b)\), then \(f\) is concave up on \((a,b)\).

• If \(f''(x) < 0\) for all \(x\) on the interval \((a,b)\), then \(f\) is concave down on \((a,b)\).

Example 1

Finding intervals of concavity from the graph of a function

The following is the graph of a continuous function that is concave down on the interval \((1,4)\) and concave up on the interval \((4,7)\).

Long Text Description

There is a horizontal x-axis with the points 1, 2, 3, 4, 5, 6, and 7 marked. There is a vertical y-axis. The graph of a function is plotted on these axes. Reading from left to right, the function goes up at a slowing pace, reaches a rounded corner, decreases at an accelerating pace, decreases at a slowing pace, reaches another rounded corner, and goes upward at an accelerating pace.

Finding Intervals of Concavity using the Second Derivative

How to Find Intervals of Concavity of a Function

• Find all values of \(x\) such that \(f''(x) = 0\) or \(f''(x)\) does not exist.

• Break up domain of \(f\) into open intervals between values found in Step 1.

• Evaluate \(f''(x)\) at one value, \(c\), from each interval, \((a,b)\), found in Step 2.

  • If \(f''(c) > 0\), then \(f\) is concave up on \((a,b)\).

  • If \(f''(c) < 0\), then \(f\) is concave down on \((a,b)\).

Example 2

Finding intervals of concavity

If the second derivative of \(f(x)\) is

\[\displaystyle f''(x) = \frac{x^2-4x}{x-6}\]

find the intervals of concavity of \(f\).

Step 1:   Find all values of   \(x\)   such that   \(f''(x) = 0\).

\[\begin{align*} f''(x) &= \frac{x(x-4)}{x-6} \end{align*}\]

which equals zero when \(x=0\) and \(x=4\).

Step 2:   Find all values of   \(x\)   such that   \(f''(x)\)   does not exist.

\(f''(x)\) does not exist when \(x=6\).

Step 3:   Perform an interval sign analysis for   \(f''\).

Break up the domain of \(f\) at each value found in Steps 1 and 2. Plug one number from each subinterval into \(f''(x)\) to determine the sign of \(f''(x)\) on each interval.

\[\begin{align*} f''(-1) &= \frac{(-1)(-1-4)}{-1-6} < 0\\ f''(1) &= \frac{(1)(1-4)}{1-6} >0\\ f''(5) &= \frac{(5)(5-4)}{5-6} <0\\ f''(7) &= \frac{(7)(7-4)}{7-6} >0 \end{align*}\]

Fig. 7 Interval analysis of \(f''(x) = \frac{x(x-4)}{x-6}\).

Long Text Description

A number line with positive and negative signs assigned to intervals, which are negative to the left of zero, positive from zero to four, negative from four to six, and positive to the right of six.

Therefore, \(f\) is concave up on the intervals \((0,4)\) and \((6,\infty)\) and concave down on the intervals \((-\infty,0)\) and \((4,6)\).