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

• Definition 5.1.3 If f is differentiable on an open interval I, then f is said to be concave up on I if is increasing on I, and f is said to be concave down on I if is said to be decreasing on I.

/f

/f

Concavity

• Definition 5.1.3 If f is differentiable on an open interval I, then f is said to be concave up on I if is increasing on I, and f is said to be concave down on I if is said to be decreasing on I.

• We already learned that where a function is increasing, its derivative is positive and where it is decreasing, its derivative is negative. Lets put that idea together with this definition. What can we say?

/f

/f

Concavity

• We were told that concave up means that is increasing. If a function is increasing, we know that its derivative is positive. So where a function is concave up, is positive.

• We were also told that concave down means that is decreasing. If a function is decreasing, we know that its derivative is negative. So where a function is concave down, is negative.

/f

//f/f

//f

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. • Locate inflection points.

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.• Locate inflection points.

________|_________

• 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

• The 2nd derivative has one zero at zero. Since there is no sign change around this zero, there are no points of inflection. This function is concave up everywhere.

_____+____|____+_____ 0

.)( 4xxf

3/ 4)( xxf 2// 12)( xxf

Inflection Points

• Inflection points mark the places on the curve y = f(x) where the rate of change of y with respect to x changes from increasing to decreasing, or vice versa.

Applications

• Suppose that water is added to the flask so that the volume increases at a constant rate with respect to the time t, and let us examine the rate at which the water level rises with respect to t. Initially the water rises at a slow rate because of the wide base. However, as the diameter of the flask narrows, the rate at which the water rises will increase until the level is at the narrow point in the neck. From that point on, the rate at which the water rises will decrease.

Homework

• Pages 275-276• 1-15 odd• 23,25,35