Extreme values: Maxima and minima

October 16, 2015 1 / 12

Motivation for Maxima and Minima

Imagine you have some model which predicts profits / cost of doingsomething / . . . .Then you probably want to find a way of deciding what choices result inthe largest (for profit) or smallest (for cost) result possible.Today’s Goal: Can calculus help us here?

Today’s outline:(1)Before we want to find best possible choices, we will want to have areason to beleive that there is a best possible choice.(2) Then we will use calculus to reduce the finding of the best choice (orworst choice) to solving some equation involving the derivative!

October 16, 2015 2 / 12

Motivation for Maxima and Minima

Imagine you have some model which predicts profits / cost of doingsomething / . . . .Then you probably want to find a way of deciding what choices result inthe largest (for profit) or smallest (for cost) result possible.Today’s Goal: Can calculus help us here?

Today’s outline:(1)Before we want to find best possible choices, we will want to have areason to beleive that there is a best possible choice.(2) Then we will use calculus to reduce the finding of the best choice (orworst choice) to solving some equation involving the derivative!

October 16, 2015 2 / 12

Hiking

Imagine you are walking and your height as a function of time is given byh(t). Suppose that the graph is given by :

Where does it look like you are at a peak? Where does it look like you arein a valley? Where on your hike are you at the greatest elevation? Whereare you at your lowest elevation?If this were a graph of expected price of a doodad over time, then when isthe best time to buy a Thneed?

October 16, 2015 3 / 12

Hiking

Imagine you are walking and your height as a function of time is given byh(t). Suppose that the graph is given by :

Where does it look like you are at a peak? Where does it look like you arein a valley? Where on your hike are you at the greatest elevation? Whereare you at your lowest elevation?

If this were a graph of expected price of a doodad over time, then when isthe best time to buy a Thneed?

October 16, 2015 3 / 12

Hiking

Imagine you are walking and your height as a function of time is given byh(t). Suppose that the graph is given by :

Where does it look like you are at a peak? Where does it look like you arein a valley? Where on your hike are you at the greatest elevation? Whereare you at your lowest elevation?If this were a graph of expected price of a doodad over time, then when isthe best time to buy a Thneed?

October 16, 2015 3 / 12

Definitions: Global maxima and minimaFor a function f , defined on an interval I containing a point c . c is calledan Absolute Maximum (or Global Maximum) if f (c) is greater and orequal to f (x) for all values of x in I .For a function f , defined on an interval I containing a point c . c is calledan Absolute Minimum (or Global Minimum) if f (c) is greater and orequal to f (x) for all values of x in I .

Where are the absolute maxima and minima for these functions. Howmany are there?

cos(x) (x − 1)2 1/x This functionon (0, 2π) on [0, 2] on (0,∞) on [0, 2]

That is weird. Not all functions have an absolute maximum / minimum.And some have many maxima / minima.

October 16, 2015 4 / 12

Definitions: Global maxima and minimaFor a function f , defined on an interval I containing a point c . c is calledan Absolute Maximum (or Global Maximum) if f (c) is greater and orequal to f (x) for all values of x in I .For a function f , defined on an interval I containing a point c . c is calledan Absolute Minimum (or Global Minimum) if f (c) is greater and orequal to f (x) for all values of x in I .Where are the absolute maxima and minima for these functions. Howmany are there?

cos(x) (x − 1)2 1/x This functionon (0, 2π) on [0, 2] on (0,∞) on [0, 2]

That is weird. Not all functions have an absolute maximum / minimum.And some have many maxima / minima.

October 16, 2015 4 / 12

Definitions: Global maxima and minimaFor a function f , defined on an interval I containing a point c . c is calledan Absolute Maximum (or Global Maximum) if f (c) is greater and orequal to f (x) for all values of x in I .For a function f , defined on an interval I containing a point c . c is calledan Absolute Minimum (or Global Minimum) if f (c) is greater and orequal to f (x) for all values of x in I .Where are the absolute maxima and minima for these functions. Howmany are there?

cos(x) (x − 1)2 1/x This functionon (0, 2π) on [0, 2] on (0,∞) on [0, 2]

That is weird. Not all functions have an absolute maximum / minimum.And some have many maxima / minima.

October 16, 2015 4 / 12

The extreme value theorem

What went wrong in these examples?

Well some of the functions just weren’t continuous. Discontinuousfunctions can do anything they want.In some examples the functions were defined on intervals that were lackingan endpoint (or the endpoint was infinite.) The function could keepincreasing (or decreasing) as it approached that endpoint.If we rule out these problems then all functions have (possibly many)maxima / minima.

Theorem (The extreme value theorem)

If a function is continuous on some closed interval, then it has anabsolute maximum and an absolute minimum on that closed interval.

October 16, 2015 5 / 12

The extreme value theorem

What went wrong in these examples?Well some of the functions just weren’t continuous. Discontinuousfunctions can do anything they want.

In some examples the functions were defined on intervals that were lackingan endpoint (or the endpoint was infinite.) The function could keepincreasing (or decreasing) as it approached that endpoint.If we rule out these problems then all functions have (possibly many)maxima / minima.

Theorem (The extreme value theorem)

If a function is continuous on some closed interval, then it has anabsolute maximum and an absolute minimum on that closed interval.

October 16, 2015 5 / 12

The extreme value theorem

What went wrong in these examples?Well some of the functions just weren’t continuous. Discontinuousfunctions can do anything they want.In some examples the functions were defined on intervals that were lackingan endpoint (or the endpoint was infinite.) The function could keepincreasing (or decreasing) as it approached that endpoint.

If we rule out these problems then all functions have (possibly many)maxima / minima.

Theorem (The extreme value theorem)

If a function is continuous on some closed interval, then it has anabsolute maximum and an absolute minimum on that closed interval.

October 16, 2015 5 / 12

The extreme value theorem

What went wrong in these examples?Well some of the functions just weren’t continuous. Discontinuousfunctions can do anything they want.In some examples the functions were defined on intervals that were lackingan endpoint (or the endpoint was infinite.) The function could keepincreasing (or decreasing) as it approached that endpoint.If we rule out these problems then all functions have (possibly many)maxima / minima.

Theorem (The extreme value theorem)

If a function is continuous on some closed interval, then it has anabsolute maximum and an absolute minimum on that closed interval.

October 16, 2015 5 / 12

The extremal value theorem

Theorem (The extremal value theorem)

If a function is continuous on some closed, bounded interval [a, b], thenit has an absolute maximum and an absolute minimum on that closedinterval.

What this theorem tells us is that looking for absolute maxima and minimais a reasonable thing to do, as long as we are working on closed intervals.If we are working on an open internal, then there are no guarantees.

Does f (x) = x + cos(x) have a absolute maximum on [−π, π]? How aboutan absolute minimum?Does is have an absolute maximum on (−∞,∞)? What about anabsolute Minimum? Try graphing

October 16, 2015 6 / 12

The extremal value theorem

Theorem (The extremal value theorem)

If a function is continuous on some closed, bounded interval [a, b], thenit has an absolute maximum and an absolute minimum on that closedinterval.

What this theorem tells us is that looking for absolute maxima and minimais a reasonable thing to do, as long as we are working on closed intervals.If we are working on an open internal, then there are no guarantees.Does f (x) = x + cos(x) have a absolute maximum on [−π, π]? How aboutan absolute minimum?

Does is have an absolute maximum on (−∞,∞)? What about anabsolute Minimum? Try graphing

October 16, 2015 6 / 12

The extremal value theorem

Theorem (The extremal value theorem)

If a function is continuous on some closed, bounded interval [a, b], thenit has an absolute maximum and an absolute minimum on that closedinterval.

What this theorem tells us is that looking for absolute maxima and minimais a reasonable thing to do, as long as we are working on closed intervals.If we are working on an open internal, then there are no guarantees.Does f (x) = x + cos(x) have a absolute maximum on [−π, π]? How aboutan absolute minimum?Does is have an absolute maximum on (−∞,∞)? What about anabsolute Minimum? Try graphing

October 16, 2015 6 / 12

Today’s goal: Finding absolute maxima and minima, nowthat we know they exist.We know that continuous functions on closed intervals have globalmaxima. Can we find them?

Calculus studies things locally. Think about the definition of the derivativefor example. Motivated by this we will “localize” the idea of a maximum.If c is a global maximum then f (c) is greater than or equal to f (x) for xin I . Then in particular f (c) is greater than or equal to f (x) for x close toc in I .

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

If c is a global maximum, then it is a local maximum. So If you find a listcontaining every local maximum then that list contains the globalmaximum? We have some plots to look at on the notes.How can calculus help us here?

October 16, 2015 7 / 12

Today’s goal: Finding absolute maxima and minima, nowthat we know they exist.We know that continuous functions on closed intervals have globalmaxima. Can we find them?Calculus studies things locally. Think about the definition of the derivativefor example. Motivated by this we will “localize” the idea of a maximum.

If c is a global maximum then f (c) is greater than or equal to f (x) for xin I . Then in particular f (c) is greater than or equal to f (x) for x close toc in I .

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

If c is a global maximum, then it is a local maximum. So If you find a listcontaining every local maximum then that list contains the globalmaximum? We have some plots to look at on the notes.How can calculus help us here?

October 16, 2015 7 / 12

Today’s goal: Finding absolute maxima and minima, nowthat we know they exist.We know that continuous functions on closed intervals have globalmaxima. Can we find them?Calculus studies things locally. Think about the definition of the derivativefor example. Motivated by this we will “localize” the idea of a maximum.If c is a global maximum then f (c) is greater than or equal to f (x) for xin I . Then in particular f (c) is greater than or equal to f (x) for x close toc in I .

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

If c is a global maximum, then it is a local maximum. So If you find a listcontaining every local maximum then that list contains the globalmaximum? We have some plots to look at on the notes.How can calculus help us here?

October 16, 2015 7 / 12

Today’s goal: Finding absolute maxima and minima, nowthat we know they exist.We know that continuous functions on closed intervals have globalmaxima. Can we find them?Calculus studies things locally. Think about the definition of the derivativefor example. Motivated by this we will “localize” the idea of a maximum.If c is a global maximum then f (c) is greater than or equal to f (x) for xin I . Then in particular f (c) is greater than or equal to f (x) for x close toc in I .

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

If c is a global maximum, then it is a local maximum. So If you find a listcontaining every local maximum then that list contains the globalmaximum? We have some plots to look at on the notes.How can calculus help us here?

October 16, 2015 7 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?

Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either

f ′(c) = 0

or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means:

If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either

f ′(c) = 0

or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximum

Similarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either

f ′(c) = 0

or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either

f ′(c) = 0

or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either

f ′(c) = 0

or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either f ′(c) = 0 or

f ′(c) does not exist

or

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either f ′(c) = 0 or f ′(c) does not existor

c is an endpoint of the interval

.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Finding local maxima and minima

Definition

For a function f , defined on an interval I containing a point c . c is calleda local Maximum if f (c) is greater and or equal to f (x) for all x nearby cin I .

How can calculus help us here?Think about what the derivative means: If f ′(c) > 0 then you are stillascending at c. If you take one more step then you will be higher. c doesnot produce a local maximumSimilarly if f ′(c) < 0 then c is not a local maximum. Taking a stepbackwards would produce greater height.

Theorem

If c is a local maximum of f then either f ′(c) = 0 or f ′(c) does not existor c is an endpoint of the interval.

Example: Find a list of candidates to be local maxima of x3 − 3x + 2 on[−1, 2]

October 16, 2015 8 / 12

Critical points

Definition (Critical points)

For a function f defined on an interval I c is a critical point if f ′(c) = 0 orf ′(c) does not exist or c is an endpoint of the interval

Notice that the points on the boundary of the interval are automaticallycritical points.The theorem on the previous slide is

Theorem

If c is a local maximum of f then c is a critical point.

Example: What are the critical points of x3 − x2 − x − 1 on [−1, 2]. Findthe global maximum. Find the global minimum.

October 16, 2015 9 / 12

A general strategy:

An Algorithm to find the global maximum of f (x) on the interval I :(1) Compute f ′(x)(2) Find the critical points of f (INCLUDING THE BOUNDARY POINTS)(3) Evaluate the function at those points and see which one is the biggest.

Example: Find the global maximum of f (x) = x3 − x2 + 1 on [−10, 10].

Example: Find the global maximum of f (x) = x4 − |x | on [−10, 10].

October 16, 2015 10 / 12

maximizing areas of rectangles

Of all rectangles which have total perimeter 50, which one has thegreatest area?Strategy:

1 In terms of length (`) and width (w), what does it mean to haveperimeter 50? Solve for ` in terms of w . (or w in terms of `.)

2 Express area in terms of ` and w . Eliminate one of these variable.

3 Since the perimeter is 50, could ` possible be greater than 50? Donegative numbers make sense for ` and w?

4 Write down the closed interval which is the domain of interest.

5 Maximize.

October 16, 2015 11 / 12

Group Work

Maximize x3 − x + 1 on [−10, 10]

Maximize f (x) =x

|x |+ 1on [−10, 10].

Maximize x + cos−1(x) on [−1, 1]

October 16, 2015 12 / 12