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

### First Derivative Test for Relative Extrema

Let *f* be a continuous function and c be a critical number of *f*. (Figure 7.2-9.)

- If
*f '*(*x*) changes from positive to negative at*x*=*c*(*f '*> 0 for*x*<*c*and*f*' < 0 for*x*>*c*), then*f*has a relative maximum at*c*. - If
*f '*(*x*) changes from negative to positive at*x*=*c*(*f '*< 0 for*x*<*c*and*f '*> 0 for*x*>*c*), then*f*has a relative minimum at*c*.

### Second Derivative Test for Relative Extrema

Let *f* be a continuous function at a number *c*.

- If
*f '*(*c*) = 0 and*f "*(*c*) < 0, then*f*(*c*) is a relative maximum. - If
*f '*(*c*) = 0 and*f "*(*c*) > 0, then*f*(*c*) is a relative minimum. - If
*f '*(*c*) = 0 and*f "*(*c*) = 0, then the test is inconclusive. Use the First Derivative Test.

### Example 1

See Figure 7.2-10. The graph of *f '*, the derivative of a function *f*, is shown in Figure 7.2-10. Find the relative extrema of *f*.

Solution: See Figure 7.2-11.

Thus *f* has a relative maximum at *x* = –2, and a relative minimum at *x* =3.

### Example 2

Find the relative extrema for the function

Step 1: Find *f '*(*x* ).

*f '*(

*x*)=

*x*

^{2}– 2

*x*– 3

Step 2: Find all critical numbers of *f* (*x*).

- Note that

*f '*(

*x*) is defined for all real numbers.

- Set

*f '*(

*x*)=0:

*x*

^{2}– 2

*x*– 3=0 (

*x*– 3)(

*x*+1)=0

*x*=3 or

*x*= – 1.

Step 3: Find *f''*(*x*): *f''*(*x*)= 2*x* – 2.

*f''*(3) = 2(3) – 2 = 4

*f*(3) is a relative minimum.

*f "*(–1)=2(–1) – 2= – 4

*f*(–1) is a relative maximum.

- Therefore, –9 is a relative minimum value of

*f*and is a relative maximum value. (See Figure 7.2-12.)

### Example 3

Find the relative extrema for the function .

**Using the First Derivative Test**

Step 1: Find *f '*(*x*).

Find all critical numbers of *f*.

- Set

*f '*(

*x*)=0. Thus 4

*x*=0 or

*x*=0.

- Set

*x*

^{2}– 1=0. Thus,

*f '*(

*x*) is undefined at

*x*=1 and

*x*= – 1. Therefore, the critical numbers are –1, 0 and 1.

Step 3: Determine intervals.

- The intervals are (–∞, –1), (–1, 0), (0, 1), and (1,∞).

Step 4: Set up a table.

Step 5: Write a conclusion.

- Using the First Derivative Test, note that

*f*(

*x*) has a relative maximum at

*x*=0 and relative minimum at

*x*= – 1 and

*x*=1.

Note that *f* (–1)=0, *f* (0)=1, and *f* (1)=0. Therefore, 1 is a relative maximum value and 0 is a relative minimum value. (See Figure 7.2-13.)

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

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