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Definition Let f be defined on an interval, and let x 1 and x 2 denote points in that interval. a)f is increasing on the interval if f(x 1 ) < f(x 2 ) whenever x 1 < x 2. b)f is decreasing on the interval if f(x 1 ) > f(x 2 ) whenever x 1 < x 2. c)f is constant on the interval if f(x 1 ) = f(x 2 ) for all points x 1 and x 2.
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Increasing/Decreasing Functions and Concavity
Objective: Use the derivative to find where a graph is increasing/decreasing
and determine concavity.
Increasing/Decreasing
• The terms increasing, decreasing, and constant are used to describe the behavior of a function over an interval as we travel left to right along its graph.
Definition 5.1.1
• Let f be defined on an interval, and let x1 and x2 denote points in that interval.
a) f is increasing on the interval if f(x1) < f(x2) whenever x1 < x2.
b) f is decreasing on the interval if f(x1) > f(x2) whenever x1 < x2.
c) f is constant on the interval if f(x1) = f(x2) for all points x1 and x2 .
The Derivative
• Lets look at a graph that is increasing. What can you tell me about the derivative of this function?
The Derivative
• Lets look at a graph that is decreasing. What can you tell me about the derivative of this function?
The Derivative
• Lets look at a graph that is constant. What can you tell me about the derivative of this function?
Theorem 5.1.2
• Let f be a function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b).
a) If for every value of x in (a, b), then f is increasing on [a, b].
b) If for every value of x in (a, b), then f is decreasing on [a, b].
c) If for every value of x in (a, b), then f is constant on [a, b]
0)(/ xf
0)(/ xf
0)(/ xf
Example 1
• Find the intervals on which is increasing and the intervals on which it is decreasing.
34)( 2 xxxf
Example 1
• Find the intervals on which is increasing and the intervals on which it is decreasing.
• We want to take the derivative and do sign analysis to see where it is positive or negative.
34)( 2 xxxf
Example 1
• Find the intervals on which is increasing and the intervals on which it is decreasing.
• We want to take the derivative and do sign analysis to see where it is positive or negative.
_________|_________
34)( 2 xxxf
42)(/ xxf
2
Example 1
• Find the intervals on which is increasing and the intervals on which it is decreasing.
• We want to take the derivative and do sign analysis to see where it is positive or negative.
_________|_________
• So increasing on , decreasing on
34)( 2 xxxf
42)(/ xxf
2
),2[ ].2,(
Example 2
• Find the intervals on which is increasing and intervals on which it is decreasing.
3)( xxf
Example 2
• Find the intervals on which is increasing and intervals on which it is decreasing.
________|________
• This function is increasing on
3)( xxf
2/ 3)( xxf
0
).,(
Example 3
• Use the graph below to make a conjecture about the intervals on which f is increasing or decreasing.
21243)( 234 xxxxf
Example 3
• Use theorem 5.1.2 to verify your conjecture.
________|______|___|________
• Increasing Decreasing
21243)( 234 xxxxf
xxxxf 241212)( 23/ )1)(2(12)(/ xxxxf
2 0 1
_ _
),1[]0,2[ and ]1,0[]2,( and
Concavity
• This graph is what we call concave up. Lets look at the derivative of this graph What is it doing? Is it increasing or decreasing?
Concavity
• This graph is what we call concave down. Lets look at the derivative of this graph What is it doing? Is it increasing or decreasing?
Concavity
• Theorem 5.1.4 Let f be twice differentiable on a open interval I.
a) If for every value of x in I, then f is concave up on I.
b) If for every value of x in I, then if is concave down on I.
0)(// xf
0)(// xf
Inflection Points
• Definition 5.1.5 If f is continuous on an open interval containing a value x0 , and if f changes the direction of its concavity at the point (x0 , f(x0)), then we say that f has an inflection point at x0 , and we call the point an inflection point of f.
Example 5
• Given , find the intervals on which f is increasing/decreasing and concave up/down. Locate all points of inflection.
13)( 23 xxxf
Example 5
• Given , find the intervals on which f is increasing/decreasing and concave up/down. Locate all points of inflection.
________|______|________
• Increasing on• Decreasing on
13)( 23 xxxf
)2(363)( 2/ xxxxxf
0 2
),2[]0,( and
]2,0[
Example 5• Given , find the intervals on which f is
increasing/decreasing and concave up/down. Locate all points of inflection.
________|______
• Concave up on• Concave down on• Inflection point at x = 1
13)( 23 xxxf
)1(666)(// xxxf
1
),1(
)1,(
Differences
• When we express where a function is increasing or decreasing, we include the points where it changes in our answers.
• Increasing• Decreasing• When we express where a function is concave up or
concave down, the inflection points are not included in the answers.
• Concave up• Concave down
),2[]0,( and
]2,0[
),1(
)1,(
Example 6
• Given• Find where this function is increasing/decreasing.• Find where this function is concave up/down.
xxexf )(
Example 6
• Given• Find where this function is increasing/decreasing.• Find where this function is concave up/down.• Locate inflection points.
________|_____
Increasing on Decreasing on
xxexf )(
)1()1()()(/ xeeexxf xxx
1
]1,( ),1[
Example 6
• Given• Find where this function is increasing/decreasing.• Find where this function is concave up/down.
________|_________
• Concave up on• Concave down on . Inflection point at x = 2.
xxexf )(
)2())(1()1()(// xeexexf xxx
2
),2(
)2,(
Example 7
• Given on the interval • Find increase/decrease.• Find concave up/down.• Inflection points.
xxxf sin2)( ]2,0[
Example 7
• Given on the interval • Find increase/decrease.• Find concave up/down.• Inflection points.
|_____|______|_____| |______|_____|
• Increasing Decreasing• C up C down Inflection point x =
xxxf sin2)(
xxf cos21)(/ xxf sin2)(//
3/2 3/4
]2,3/4[]3/2,0[ and ]3/4,3/2[
)2,( ),0(
]2,0[
0 02 2
Example 8
• Find the inflection points, if any, of
• This function is concave up everywhere.
.)( 4xxf
3/ 4)( xxf 2// 12)( xxf
Homework
• Pages 275-276• 1-15 odd• 23,25,35
