1. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
OBJECTIVES
Find relative extrema of a continuous function using the First-Derivative Test.
Shi,Chen

2. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
DEFINITIONS:
A function f is increasing over I if, for every a and b
in I, if a < b, then f (a) < f (b).
(If the input a is less than the input b, then the output
for a is less than the output for b.
A function f is decreasing over I if, for every a and b
in I, if a < b, then f (a) > f (b).
for a is greater than the output for b.)

3. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right.
dec
inc
inc

4. The slope of the tan line is positive when the function is increasing and negative when decreasing

5. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
THEOREM 1
If f(x) > 0 for all x in an interval I, then f is increasing over I.
If f(x) < 0 for all x in an interval I, then f is decreasing over I.

6. Find the intervals where f is increasing and decreasing

7. Find the intervals where f is increasing and decreasing
Since f ’(x) = 2x+5 it follows that
f is increasing when 2x+5>0 or
when x>-2.5 which is the interval

8. We use a similar method to find the interval where f is decreasing
We use a similar method to find the interval where f is decreasing. 2x+5<0 gives x < -2.5 or the interval
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

9. Find the intervals where the function is increasing and decreasing

10. A product has a profit function of for the production and sale of x units. Is the profit increasing or decreasing when 100 units have been sold?

11. Suppose a product has a cost function given by Find the average cost function. Over what interval is the average cost decreasing?

12. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
DEFINITION:
A critical value of a function f is any number c in the
domain of f for which the tangent line at (c, f (c)) is
horizontal or for which the derivative does not exist.
That is, c is a critical value if f (c) exists and
f (c) = 0 or f (c) does not exist.
p. 200 The definition of critical value has a period at the end of the first c and a capital “in the domain…” . The grammar here makes no sense.

13. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
DEFINITIONS:
Let I be the domain of f :
f (c) is a relative minimum if there exists within I an
open interval I1 containing c such that f (c) ≤ f (x) for
all x in I1;
and
F (c) is a relative maximum if there exists within I an
open interval I2 containing c such that f (c) ≥ f (x) for
all x in I2.

14. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
THEOREM 2
If a function f has a relative extreme value f (c) on an
open interval; then c is a critical value. So,
f (c) = 0 or f (c) does not exist.

15. THEOREM 3: The First-Derivative Test for Relative Extrema
Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
THEOREM 3: The First-Derivative Test for
Relative Extrema
For any continuous function f that has exactly one
critical value c in an open interval (a, b);
F1. f has a relative minimum at c if f (x) < 0 on
(a, c) and f (x) > 0 on (c, b). That is, f is decreasing to the left of c and increasing to the right of c.

16. THEOREM 3: The First-Derivative Test for Relative Extrema (continued)
Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
THEOREM 3: The First-Derivative Test for
Relative Extrema (continued)
F2. f has a relative maximum at c if f (x) > 0 on
(a, c) and f (x) < 0 on (c, b). That is, f is increasing to the left of c and decreasing to the right of c.
F3. f has neither a relative maximum nor a relative minimum at c if f (x) has the same sign on (a, c) and (c, b).

17. Example 1: Graph the function f given by
Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1: Graph the function f given by
and find the relative extrema.
Suppose that we are trying to graph this function but
do not know any calculus. What can we do? We can
plot a few points to determine in which direction the
graph seems to be turning. Let’s pick some x-values
and see what happens.

18. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1 (continued):

19. 2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1 (continued):
We can see some features of the graph from the sketch.
Now we will calculate the coordinates of these features
precisely.
1st find a general expression for the derivative.
2nd determine where f (x) does not exist or where
f (x) = 0. (Since f (x) is a polynomial, there is no
value where f (x) does not exist. So, the only
possibilities for critical values are where f (x) = 0.)

20. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1 (continued):
These two critical values partition the number line into
3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

21. f is increasing on (–∞, –1]
Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1 (continued):
3rd analyze the sign of f (x) in each interval.
Test Value
x = –2
x = 0
x = 4
Sign of
f (x) + –
Result
f is increasing on (–∞, –1]
f is decreasing on [–1, 2]
f is increasing on [2, ∞)

22. 2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Example 1 (concluded):
Therefore, by the First-Derivative Test,
f has a relative maximum at x = –1 given by
Thus, (–1, 19) is a relative maximum.
And f has a relative minimum at x = 2 given by
Thus, (2, –8) is a relative minimum.