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2.1
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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
OBJECTIVE• Find relative extrema of a continuous function using the First-Derivative Test.
• Sketch graphs of continuous functions.
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DEFINITION:A function f is increasing over I if, for every a and b in I, if a < b, then f (a) < f (b).
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
(If the input a is less than the input b, then the output for a is less than the output for b.
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DEFINITION:A function f is decreasing over I if, for every a and b in I, if a < b, then f (a) > f (b).
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
(If the input a is less than the input b, then the output for a is greater than the output for b.)
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THEOREM 1
Let f be differentiable over an open interval I.
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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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).
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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Graph over the interval (a, b)
f (c) Sign of f (x) for x in (a, c)
Sign of f (x) for x in (c, b)
Increasing or decreasing
Relative minimum
Relative maximum
–
+
+
–
Decreasing on (a, c];
increasing on [c, b)
Increasing on (a, c];
decreasing on [c, b)
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Graph over the interval (a, b)
f (c) Sign of f (x) for x in (a, c)
Sign of f (x) for x in (c, b)
Increasing or decreasing
No relative maxima
or minima
No relative maxima
or minima
–
+
–
+
Decreasing on (a, b)
Increasing on (a, b)
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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-valuesand see what happens.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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Example 1 (continued):
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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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.)
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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Example 1 (continued):
These two critical values partition the number line into 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).
26 6 12 0x x
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
2 1x or x
( 2)( 1) 0x x
26 6 12 0x x
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Example 1 (continued):3rd analyze the sign of f (x) in each interval.
Test Value x = –2 x = 0 x = 4
Sign off (x)
Result
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
+ – +
f is increasing on (–∞, –1]
f is decreasing on [–1, 2]
f is increasing on [2, ∞)
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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.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 1
Graph the function given by , and find the relative extrema.
Relative Maximum at:
Relative Minimum at:
g 3( ) 27 6g x x x
3,48
3, 60
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Example 2: Find the relative extrema for the Function f (x) given by
Then sketch the graph.
1st find f (x).
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
3
2( )
3 2f x
x
1 32
( ) 23
f x x
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Example 2 (continued): Then, find where f (x) does not exist or where f (x) = 0.
Note that f (x) does not exist where the denominator equals 0. Since the denominator equals 0 when x = 2, x = 2 is a critical value.
f (x) = 0 where the numerator equals 0. Since 2 ≠ 0, f (x) = 0 has no solution.Thus, x = 2 is the only critical value.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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Example 2 (continued): 3rd x = 2 partitions the number line into 2 intervals:
A (– ∞, 2) and B (2, ∞). So, analyze the signs of f (x) in both intervals.
Test Value x = 0 x = 3
Sign of f (x)
Result
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
– +
f is decreasing on (– ∞, 2]
f is increasing on [2, ∞)
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Example 2 (continued):Therefore, by the First-Derivative Test,
f has a relative minimum at x = 2 given by
Thus, (2, 1) is a relative minimum.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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Example 2 (concluded):We use the information obtained to sketch the graph below, plotting other function values as needed.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 2
Find the relative extrema of the function given by . Then sketch the graph.
First find :
h 4 38( )
3h x x x
( )h x
3 284 3
3h x x x
3 24 8h x x x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 2 Continued
Second find where does not exist or where .
So when and .
Thus and are the critical values.
h x 0h x
3 24 8 0h x x x 24 2 0x x
0h x 0x 2x
0x 2x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 2 Continued
Third, and partitions the number line into three intervals: A , B , and C . So analyze the signs of in all three intervals.
Thus, there is a minimum at . Therefore, is a minimum.
0x 2x ,0 0,2 2, h x
Interval
Test Value
Sign of
Result
A B C
2x 1x 3x
h x
is decreasingh is decreasingh is increasingh
2x 16
2,3
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 2 Concluded
From the information we have gathered, the graph of looks like:
h x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 3
Find the relative extrema of the function given by . Then sketch the graph.
First find :
g 1/3( ) 3g x x
g x
1
131
3g x x
2
3
1
3
g x
x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 3 Continued
Second, find where does not exist or where .
Note: does not exist when the denominator = 0. So does not exist when . Also, there is no value of that makes
.
Thus there is a critical value at .
g x 0g x
g x g x0x x
0g x
0x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 3 Continued
Third, partitions the number line into two intervals: and . So analyze the signs of for both intervals.
Thus there is no extrema for .
0x ,0A 0,B g x
Interval
Test Value
Sign of
Results
,0A 0,B
1x 1x
g x
is increasingg is increasingg
g x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Quick Check 3 Concluded
Using the information gathered, the graph of looks like: g x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Section Summary
• A function is increasing over an interval I if, for all a and b in I
such that a < b, we have f(a) < f(b). Equivalently, the slope of the
secant line connecting a and b is positive:
• A function is decreasing over an interval I if, for all a and b in I
such that a < b, we have f(a) > f(b). Equivalently, the slope of the
secant line connecting a and b is negative:
f
f
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Section Summary Continued
• A function is increasing over an open interval I if, for all x in I, the slope of the tangent line at x is positive; that is, f(x) > 0. Similarly, a function is decreasing over an open interval I if, for all x in I, the slope of the tangent line is negative; that is f(x) < 0.
• A critical value is a number c in the domain of f such that f(c) = 0 or f(c) does not exist. The point (c, f(c)) is called a critical point.
• A relative maximum point is higher than all other points in some interval containing it. Similarly, a relative minimum point is lower than all other points in some interval containing it. They y-value of such a point is called a relative maximum (or minimum) value of the function.
x
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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
Section Summary Concluded
• Minimum and maximum points are collectively called extrema.
• Critical values are candidates for possible relative extrema. The First-Derivative Test is used to classify a critcal value as a relative minimum, a relative maximum, or neither.