Increasing and Decreasing Functions and the First Derivative Test
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Increasing and Decreasing Functions and the First Derivative Test
Determine the intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function
Standard 4.5a
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Incr
easin
g
Constant
Decreasing
y
x
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Test for Increasing and Decreasing Functions
Let f be differentiable on the interval (a, b)
1. If f’(x) > 0 then f is increasing on (a, b)2. If f’(x) < 0 then f is decreasing on (a, b)3. If f’(x) = 0 then f is constant on (a, b)
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Definition of Critical Number
If f is defined at c, then c is a critical number of f if f’(c)=0 or if f’ is undefined at c.
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Find the open intervals on which the given function is increasing or decreasing.
1. Find derivative.
2. Set f’(x) = 0 and solve to find the critical numbers.
CRITICAL NUMBERS
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3. Make table to test the sign f’(x) in each interval.
4. Use the test for increasing/decreasing to decide whether f is increasing or decreasing on each interval.
Interval -∞ < x < -2 -2 < x < 2 2 < x < ∞
Test value x = -3 x = 0 x = 3
Sign of f’(x) f’(-3) > 0 f’(0) < 0 f’(3) > 0
Conclusion Increasing Decreasing Increasing
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Find the open intervals on which the given function is increasing or decreasing.
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y
x
Incr
easin
g Decreasing
Incr
easin
gRelative maximum
Relative minimum
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Definition of Relative Extrema
Let f be a function defined at c.
1. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b).
2. f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b).
If f(c) is a relative extremum of f, then the relative extremum is said to occur at x = c.
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1. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b).
c
relative maximum f(c)
f(x)
f(x)
f(x)
f(x) f(x)f(x)
f(x)
f(x)
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2. f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b).
relative minimum f(c)
f(x)f(x)
f(x)
f(x)f(x)
f(x)
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Occurrence of Relative Extrema
If f has a relative minimum or a relative maximum when x = c, then c is a critical number of f. That is, either f’(c) = 0 or f’(c) is undefined.
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First-Derivative Test for Relative ExtremaLet f be continuous on the interval (a, b) in which c is the only critical number.
On the interval (a, b) if1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum.2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum.3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.
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1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum.
Relative minimum
f’(x) is positiv
ef’(x) is
negative
c
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2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum.
relative maximum
f’(x) is positiv
e
f’(x) is negative
c
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3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.
c
f’(x) is positive
f’(x) is positive
Not a relative extremum
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Find all relative extrema of the given function.
Find derivative
Set = 0 to find critical numbers
CRITICAL NUMBERS
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(-∞, -1) (-1, 1) (1, ∞)x = -2 x = 0 x = 2
+ - +Increasing Decreasing Increasing
Relative Maximum (-1, 5)
Relative Minimum (1, -3)
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Find all relative extrema of the given function.
(-∞, -2) (-2, 0) (0, ∞)x = -3 x = -1 x = 1
+ - +Increasing Decreasing Increasing
Relative max: (-2, 0) Relative min: (0, -2)
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Find all relative extrema of the given function.
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(0,π/4) (π/4,3π,4) (3π/4,5π/4) (5π/4,7π/4) (7π/4,2π)
x = π/6 x = π/2 x = π x = 3π/2 x = 2π
+ - + - +
Increasing Decreasing Increasing Decreasing Increasing
Relative max:
Relative min:
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The graph of f is shown. Sketch a graph of the derivative of f.
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The graph of f is shown. Sketch a graph of the derivative of f.
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The graph of f is shown. Sketch a graph of the derivative of f.