First Derivative Test and Second Derivative Test for Relative Extremas for AP Calculus

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.)

1. 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.
2. 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.

1. If f '(c) = 0 and f "(c) < 0, then f (c) is a relative maximum.
2. If f '(c) = 0 and f "(c) > 0, then f (c) is a relative minimum.
3. 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 )=x² – 2x – 3

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

Note that f '(x ) is defined for all real numbers.

Set f '(x )=0: x² – 2x – 3=0 (x – 3)(x +1)=0 x =3 or x = – 1.

Step 3: Find f''(x): f''(x)= 2x – 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 4x =0 or x =0.

Set x² – 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