Practice problems for these concepts can be found at: Graphs of Functions and Derivatives Practice Problems for AP Calculus

The functions *f*, *f* ', and *f*" are interrelated, and so are their graphs. Therefore, you can usually infer from the graph of one of the three functions ( *f*, *f* ', or *f* ") and obtain information about the other two. Here are some examples.

### Example 1

The graph of a function *f* is shown in Figure 7.4-1. Which of the following is true for *f* on (*a*, *b*)?

*f*' ≥ 0 on (*a*,*b*)*f*" > 0 on (*a*,*b*)

Solution:

- Since f is strictly increasing,
*f*' ≥ 0 on (*a*,*b*) is true. - The graph is concave downward on (
*a*, 0) and upward on (0,*b*). Thus*f*" > 0 on (0,*b*) only. Therefore only statement I is true.

### Example 2

Given the graph of *f* ' in Figure 7.4-2, find where the function *f* (a) has its relative maximum(s) or relative minimums, (b) is increasing or decreasing, (c) has its point(s) of inflection, (d) is concave upward or downward, and (e) if *f*(–2)= *f*(2)=1 and *f*(0)= – 3, draw a sketch of *f*.

The function f has a relative maximum at *x* = –4 and at *x* = 4, and a relative minimum at *x* = 0.

A change of concavity occurs at *x* = – 2 and at *x* = 2 and *f* ' exists at *x* = – 2 and at *x* =2, which implies that there is a tangent line to the graph of *f* at *x* = – 2 and at *x* =2. Therefore, *f* has a point of inflection at *x* = – 2 and at *x* =2.

- Summarize the information of
*f*' on a number line: - The function
*f*is increasing on interval (–∞, –4] and [0, 4], and*f*is decreasing on [–4, 0] and [4,∞). - Summarize the information of
*f*" on a number line: - The graph of
*f*is concave upward on the interval (–2, 2) and concave downward on (–∞, –2) and (2,∞). - A sketch of the graph of
*f*is shown in Figure 7.4-3.

### Example 3

Given the graph of *f* ' in Figure 7.4-4, find where the function *f* (a) has a horizontal tangent, (b) has its relative extrema, (c) is increasing or decreasing, (d) has a point of inflection, and (e) is concave upward or downward.

The First Derivative Test indicates that *f* has relative maximums at *x* = – 4 and 4; and *f* has relative minimums at *x* =2 and 8.

A change of concavity occurs at *x* = – 1, 3, and 6. Since *f* '(x ) exists, *f* has a tangent at every point. Therefore, *f* has a point of inflection at *x* = – 1, 3, and 6.

*f*' (*x*)=0 at*x*= – 4, 2, 4, 8. Thus*f*has a horizontal tangent at these values.- Summarize the information of
*f*' on a number line: - The function
*f*is increasing on (–8, –4], [2, 4], and [8, ∞) and is decreasing on [–4, 2] and [4, 8]. - Summarize the information of
*f*" on a number line: - The function
*f*is concave upward on (–1, 3) and (6, ∞) and concave downward on (–∞, –1) and (3, 6).

### Example 4

A function *f* is continuous on the interval [–4, 3] with *f*(–4)=6 and *f*(3)=2 and the following properties:

- Find the intervals on which
*f*is increasing or decreasing. - Find where
*f*has its absolute extrema. - Find where
*f*has the points of inflection. - Find the intervals where
*f*is concave upward or downward. - Sketch a possible graph of
*f*.

Solution:

- The graph of f is increasing on [1, 3] and decreasing on [–4, –2] and [–2, 1].
- At
*x*= – 4,*f*(*x*)=6. The function decreases until*x*=1 and increases back to 2 at*x*=3. Thus, f has its absolute maximum at*x*= – 4 and its absolute minimum at*x*=1. - A change of concavity occurs at
*x*= –2, and since*f*'(–2)=0 which implies a tangent line exists at*x*= – 2, f has a point of inflection at*x*= – 2. - The graph of
*f*is concave upward on (–4, –2) and concave downward on (–2, 1) and (1, 3). - A possible sketch of
*f*is shown in Figure 7.4-5.

### Example 5

If *f *(*x* )= |ln(*x* +1)|, find *f* '(*x*). (See Figure 7.4-6.)

The domain of *f* is (–1,∞).

*f* (0) = |ln(0+1)| = |ln(1)| = 0

Practice problems for these concepts can be found at:

Graphs of Functions and Derivatives Practice Problems for AP Calculus

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