[Update] Minimum and Maximum Values | minimum – POLLICELEE

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Section 4-3 : Minimum and Maximum Values

Many of our applications in this chapter will revolve around minimum and maximum values of a function. While we can all visualize the minimum and maximum values of a function we want to be a little more specific in our work here. In particular, we want to differentiate between two types of minimum or maximum values. The following definition gives the types of minimums and/or maximums values that we’ll be looking at.


  1. We say that \(f\left( x \right)\) has an absolute (or global) maximum at \(x = c\) if\(f\left( x \right) \le f\left( c \right)\) for every \(x\) in the domain we are working on.
  2. We say that \(f\left( x \right)\) has a relative (or local) maximum at \(x = c\) if \(f\left( x \right) \le f\left( c \right)\) for every \(x\) in some open interval around \(x = c\).
  3. We say that \(f\left( x \right)\) has an absolute (or global) minimum at \(x = c\) if \(f\left( x \right) \ge f\left( c \right)\) for every \(x\) in the domain we are working on.
  4. We say that \(f\left( x \right)\) has a relative (or local) minimum at \(x = c\) if\(f\left( x \right) \ge f\left( c \right)\) for every \(x\) in some open interval around \(x = c\).

Note that when we say an “open interval around\(x = c\)” we mean that we can find some interval \(\left( {a,b} \right)\), not including the endpoints, such that \(a < c < b\). Or, in other words, \(c\) will be contained somewhere inside the interval and will not be either of the endpoints.

Also, we will collectively call the minimum and maximum points of a function the extrema of the function. So, relative extrema will refer to the relative minimums and maximums while absolute extrema refer to the absolute minimums and maximums.

Now, let’s talk a little bit about the subtle difference between the absolute and relative in the definition above.

We will have an absolute maximum (or minimum) at \(x = c\) provided \(f\left( c \right)\) is the largest (or smallest) value that the function will ever take on the domain that we are working on. Also, when we say the “domain we are working on” this simply means the range of \(x\)’s that we have chosen to work with for a given problem. There may be other values of \(x\) that we can actually plug into the function but have excluded them for some reason.

A relative maximum or minimum is slightly different. All that’s required for a point to be a relative maximum or minimum is for that point to be a maximum or minimum in some interval of \(x\)’s around \(x = c\). There may be larger or smaller values of the function at some other place, but relative to \(x = c\), or local to \(x = c\), \(f\left( c \right)\) is larger or smaller than all the other function values that are near it.

Note as well that in order for a point to be a relative extrema we must be able to look at function values on both sides of \(x = c\) to see if it really is a maximum or minimum at that point. This means that relative extrema do not occur at the end points of a domain. They can only occur interior to the domain.

There is actually some debate on the preceding point. Some folks do feel that relative extrema can occur on the end points of a domain. However, in this class we will be using the definition that says that they can’t occur at the end points of a domain. This will be discussed in a little more detail at the end of the section once we have a relevant fact taken care of.

It’s usually easier to get a feel for the definitions by taking a quick look at a graph.

For the function shown in this graph we have relative maximums at \(x = b\) and \(x = d\). Both of these points are relative maximums since they are interior to the domain shown and are the largest point on the graph in some interval around the point. We also have a relative minimum at \(x = c\) since this point is interior to the domain and is the lowest point on the graph in an interval around it. The far-right end point, \(x = e\), will not be a relative minimum since it is an end point.

The function will have an absolute maximum at \(x = d\) and an absolute minimum at \(x = a\). These two points are the largest and smallest that the function will ever be. We can also notice that the absolute extrema for a function will occur at either the endpoints of the domain or at relative extrema. We will use this idea in later sections so it’s more important than it might seem at the present time.

Let’s take a quick look at some examples to make sure that we have the definitions of absolute extrema and relative extrema straight.

Example 1

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = {x^2}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 1,2} \right]\]

Since this function is easy enough to graph let’s do that. However, we only want the graph on the interval \(\left[ { – 1,2} \right]\). Here is the graph,

Note that we used dots at the end of the graph to remind us that the graph ends at these points.

We can now identify the extrema from the graph. It looks like we’ve got a relative and absolute minimum of zero at \(x = 0\) and an absolute maximum of four at \(x = 2\). Note that \(x = – 1\) is not a relative maximum since it is at the end point of the interval.

This function doesn’t have any relative maximums.

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = {x^2}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 1,2} \right]\]

As we saw in the previous example functions do not have to have relative extrema. It is completely possible for a function to not have a relative maximum and/or a relative minimum.

Example 2

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = {x^2}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 2,2} \right]\]

Here is the graph for this function.

In this case we still have a relative and absolute minimum of zero at \(x = 0\). We also still have an absolute maximum of four. However, unlike the first example this will occur at two points, \(x = – 2\) and \(x = 2\).

Again, the function doesn’t have any relative maximums.

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = {x^2}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 2,2} \right]\]

As this example has shown there can only be a single absolute maximum or absolute minimum value, but they can occur at more than one place in the domain.

Example 3

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = {x^2}\]

In this case we’ve given no domain and so the assumption is that we will take the largest possible domain. For this function that means all the real numbers. Here is the graph.

In this case the graph doesn’t stop increasing at either end and so there are no maximums of any kind for this function. No matter which point we pick on the graph there will be points both larger and smaller than it on either side so we can’t have any maximums (of any kind, relative or absolute) in a graph.

We still have a relative and absolute minimum value of zero at \(x = 0\).

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = {x^2}\]

So, some graphs can have minimums but not maximums. Likewise, a graph could have maximums but not minimums.

Example 4

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = {x^3}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 2,2} \right]\]

Here is the graph for this function.

This function has an absolute maximum of eight at \(x = 2\) and an absolute minimum of negative eight at \(x = – 2\). This function has no relative extrema.

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = {x^3}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ { – 2,2} \right]\]

So, a function doesn’t have to have relative extrema as this example has shown.

Example 5

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = {x^3}\]

Again, we aren’t restricting the domain this time so here’s the graph.

In this case the function has no relative extrema and no absolute extrema.

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = {x^3}\]

As we’ve seen in the previous example functions don’t have to have any kind of extrema, relative or absolute.

Example 6

Identify the absolute extrema and relative extrema for the following function.

\[f\left( x \right) = \cos \left( x \right)\]

We’ve not restricted the domain for this function. Here is the graph.

Cosine has extrema (relative and absolute) that occur at many points. Cosine has both relative and absolute maximums of 1 at

\[x = \ldots – 4\pi ,\, – 2\pi ,\,\,0,\,\,2\pi ,\,\,4\pi , \ldots \]

\[x = \ldots – 4\pi ,\, – 2\pi ,\,\,0,\,\,2\pi ,\,\,4\pi , \ldots \]

Cosine also has both relative and absolute minimums of -1 at

\[x = \ldots – 3\pi ,\, – \pi ,\,\,\pi ,\,\,3\pi , \ldots \]

\[x = \ldots – 3\pi ,\, – \pi ,\,\,\pi ,\,\,3\pi , \ldots \]

Identify the absolute extrema and relative extrema for the following function. \[f\left( x \right) = \cos \left( x \right)\]

As this example has shown a graph can in fact have extrema occurring at a large number (infinite in this case) of points.

We’ve now worked quite a few examples and we can use these examples to see a nice fact about absolute extrema. First let’s notice that all the functions above were continuous functions. Next notice that every time we restricted the domain to a closed interval (i.e. the interval contains its end points) we got absolute maximums and absolute minimums. Finally, in only one of the three examples in which we did not restrict the domain did we get both an absolute maximum and an absolute minimum.

These observations lead us the following theorem.

Extreme Value Theorem

Suppose that \(f\left( x \right)\) is continuous on the interval \(\left[ {a,b} \right]\) then there are two numbers \(a \le c,d \le b\) so that \(f\left( c \right)\) is an absolute maximum for the function and \(f\left( d \right)\) is an absolute minimum for the function.

So, if we have a continuous function on an interval \(\left[ {a,b} \right]\) then we are guaranteed to have both an absolute maximum and an absolute minimum for the function somewhere in the interval. The theorem doesn’t tell us where they will occur or if they will occur more than once, but at least it tells us that they do exist somewhere. Sometimes, all that we need to know is that they do exist.

This theorem doesn’t say anything about absolute extrema if we aren’t working on an interval. We saw examples of functions above that had both absolute extrema, one absolute extrema, and no absolute extrema when we didn’t restrict ourselves down to an interval.

The requirement that a function be continuous is also required in order for us to use the theorem. Consider the case of

\[f\left( x \right) = \frac{1}{{{x^2}}}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}[ – 1,1]\]

\[f\left( x \right) = \frac{1}{{{x^2}}}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}[ – 1,1]\]

Here’s the graph.

This function is not continuous at \(x = 0\) as we move in towards zero the function is approaching infinity. So, the function does not have an absolute maximum. Note that it does have an absolute minimum however. In fact the absolute minimum occurs twice at both \(x = – 1\) and \(x = 1\).

If we changed the interval a little to say,

\[f\left( x \right) = \frac{1}{{{x^2}}}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ {\frac{1}{2},1} \right]\]

\[f\left( x \right) = \frac{1}{{{x^2}}}\hspace{0.25in}{\mbox{on}}\hspace{0.25in}\left[ {\frac{1}{2},1} \right]\]

the function would now have both absolute extrema. We may only run into problems if the interval contains the point of discontinuity. If it doesn’t then the theorem will hold.

We should also point out that just because a function is not continuous at a point that doesn’t mean that it won’t have both absolute extrema in an interval that contains that point. Below is the graph of a function that is not continuous at a point in the given interval and yet has both absolute extrema.

This graph is not continuous at \(x = c\), yet it does have both an absolute maximum (\(x = b\)) and an absolute minimum (\(x = c\)). Also note that, in this case one of the absolute extrema occurred at the point of discontinuity, but it doesn’t need to. The absolute minimum could just have easily been at the other end point or at some other point interior to the region. The point here is that this graph is not continuous and yet does have both absolute extrema

The point of all this is that we need to be careful to only use the Extreme Value Theorem when the conditions of the theorem are met and not misinterpret the results if the conditions aren’t met.

In order to use the Extreme Value Theorem we must have an interval that includes its endpoints, often called a closed interval, and the function must be continuous on that interval. If we don’t have a closed interval and/or the function isn’t continuous on the interval then the function may or may not have absolute extrema.

We need to discuss one final topic in this section before moving on to the first major application of the derivative that we’re going to be looking at in this chapter.

Fermat’s Theorem

If \(f\left( x \right)\) has a relative extrema at \(x = c\) and \(f’\left( c \right)\) exists then \(x = c\) is a critical point of \(f\left( x \right)\). In fact, it will be a critical point such that \(f’\left( c \right) = 0\).

To see the proof of this theorem see the Proofs From Derivative Applications section of the Extras chapter.

Also note that we can say that \(f’\left( c \right) = 0\) because we are also assuming that \(f’\left( c \right)\) exists.

This theorem tells us that there is a nice relationship between relative extrema and critical points. In fact, it will allow us to get a list of all possible relative extrema. Since a relative extrema must be a critical point the list of all critical points will give us a list of all possible relative extrema.

Consider the case of \(f\left( x \right) = {x^2}\). We saw that this function had a relative minimum at \(x = 0\) in several earlier examples. So according to Fermat’s theorem \(x = 0\) should be a critical point. The derivative of the function is,

\[f’\left( x \right) = 2x\]

\[f’\left( x \right) = 2x\]

Sure enough \(x = 0\) is a critical point.

Be careful not to misuse this theorem. It doesn’t say that a critical point will be a relative extrema. To see this, consider the following case.

\[f\left( x \right) = {x^3}\hspace{0.25in}\hspace{0.25in}f’\left( x \right) = 3{x^2}\]

\[f\left( x \right) = {x^3}\hspace{0.25in}\hspace{0.25in}f’\left( x \right) = 3{x^2}\]

Clearly \(x = 0\) is a critical point. However, we saw in an earlier example this function has no relative extrema of any kind. So, critical points do not have to be relative extrema.

Also note that this theorem says nothing about absolute extrema. An absolute extrema may or may not be a critical point.

Before we leave this section we need to discuss a couple of issues.

First, Fermat’s Theorem only works for critical points in which \(f’\left( c \right) = 0\). This does not, however, mean that relative extrema won’t occur at critical points where the derivative does not exist. To see this consider \(f\left( x \right) = \left| x \right|\). This function clearly has a relative minimum at \(x = 0\) and yet in a previous section we showed in an example that \(f’\left( 0 \right)\) does not exist.

What this all means is that if we want to locate relative extrema all we really need to do is look at the critical points as those are the places where relative extrema may exist.

Finally, recall that at that start of the section we stated that relative extrema will not exist at endpoints of the interval we are looking at. The reason for this is that if we allowed relative extrema to occur there it may well (and in fact most of the time) violate Fermat’s Theorem. There is no reason to expect end points of intervals to be critical points of any kind. Therefore, we do not allow relative extrema to exist at the endpoints of intervals.

[NEW] $15 minimum wage debate: Everything you need to know | minimum – POLLICELEE

(CNN) House Democrats have included a minimum wage hike in their latest Covid relief bill , although a ruling from the Senate parliamentarian means it can’t be included in the final package.

Improving worker pay is unrelated to pandemic relief in that raising the minimum wage has long been a Democratic priority. It is completely related in that the Covid economy has hurt low wage workers — those in food service, child care and home health care, for instance, who can’t work from home — more than it’s hurt people who are able to work from home.

The argument about income inequality, which is growing in the US , is central to the widespread effort to raise the minimum wage.

“A $15 minimum wage is not a radical idea,” said Vermont Sen. Bernie Sanders on Twitter recently. “What’s radical is the fact that millions of Americans are forced to work for starvation wages, while 650 billionaires became over $1 trillion richer during a global pandemic. Yes. We must raise the minimum wage to a living wage.”

Here’s what you need to know to understand the debate.

What is the current minimum wage?

The federally mandated wage floor is $7.25 per hour.

But the minimum wage is actually different around the country. Some states have a much higher minimum wage than others, and some cities are even higher than the rest of their states. The highest minimum wage is $15 in Washington, DC.

There’s also a big exception to the minimum wage for workers who make tips, such as waiters. The federal minimum for those workers is just $2.13 an hour.

When was the last time the minimum wage changed?

While states and cities have recently raised their own minimum wages — in some places, after voters approved ballot measures — it has been more than a decade since the federal minimum wage was last raised in 2009.

That 12 year gap is the longest American workers have ever gone without a bump.

The minimum wage was first enacted in 1938 at 25 cents an hour.

How far does that $7.25 go?

Not as far as it went in 2009! Because of inflation, you would need $8.81 per hour in 2021 dollars to equal $7.25 in 2009. In that regard, the federal minimum wage has dropped.

When was the minimum wage highest?

In terms of inflation and purchasing power, the $1.60 minimum wage in 1968 would be worth about $12.27 in 2021 dollars.

How high do Democrats want to raise the minimum wage?

Sanders wants to raise the minimum wage incrementally — by $1 or $1.50 each year — until it reaches $15 per hour in 2025.

But this next part is important. Rather than rely on Congress to agree on raising the minimum wage again in 10 years or so, a bill introduced by Sanders would require it to be raised each year by the Department of Labor based on Bureau of Labor Statistics data.

How many people make the federal minimum wage?

Because so many states have their own, higher minimum wage, relatively few American workers make the federal minimum wage.

According to BLS surveys, about 392,000 workers earned the minimum wage of $7.25 in 2019 and 1.2 million workers were paid a wage below the federal minimum. They represent 1.9% of the 82.3 million American workers paid by the hour.

But almost all workers make less than $15, which is the minimum in only a few places.

How much of the country has imposed a minimum wage higher than the federal floor?

More than half of states — 29 — have a minimum wage higher than $7.25 and 45 cities have a minimum wage higher than their state, according to the Economic Policy Institute . A large portion of the states with a minimum wage higher than $7.25 raise it automatically with inflation.

The wage went up in 20 states January 1 . California has the highest statewide minimum wage at $14 per hour for companies that employ 26 or more people — but San Francisco and other cities in the state have higher wages, as do New York, Seattle and others.

What do we know about people making minimum wage?

Again, according to BLS , they are mostly young and they are less likely to have more than a high school education, if that. They are more likely to be working part-time than full-time. A larger portion are women than men, and they are less likely to married.

The top industry for people making minimum wage is the service industry, particularly food service.

The states with the largest proportion of minimum wage workers are in the South — Louisiana, Mississippi and South Carolina, according to BLS. In the vast majority of states, less than 5% of workers paid by the hour make minimum wage.

Can you live on the minimum wage?

An individual working 40 hours a week and making the federal minimum wage would make $15,080.

The federal poverty level in 2021 is $12,880 . So a single person working full time and making minimum wage is technically above the poverty level. However, if the person has a dependent and is making minimum wage, they fall below the federal poverty level of $17,420.

The poverty level seems impossibly low.

There is criticism of the poverty level as a measure, since it does not take into account the cost of living in different places. MIT professor Amy Glasmeier has developed a “living wage” calculator , that varies by state, county and city.

It also takes into consideration different family scenarios — such as one adult with one or more children, and two adults with one or both working — and is invariably higher than even the proposed $15 minimum wage.

What would be the effect of raising the minimum wage on businesses?

Many business owners and some economists argue that raising the federal minimum wage hurts businesses and costs people jobs, particularly in parts of the country with lower cost of living.

“Large cities with high costs of living — many of which already have or are on the path to a $15 minimum wage — may not experience huge consequences. But non-urban areas and places with lower costs of living could be devastated,” writes Rachel Greszler , a research fellow at the Heritage Foundation.

What would be the effect of raising the minimum wage on low-wage earners?

The non-partisan Congressional Budget Office predicted the incremental wage hike to $15 would raise worker pay by $333 billion for 17 million workers whose pay is at or below the minimum and could also affect 10 million workers whose pay is slightly above. Nearly 1 million people would be pulled above the poverty level and the government would spend less money on food assistance programs.

For an extremely in-depth look at the benefits of raising the minimum wage, here’s a report by the Economic Policy Institute , which lobbies for a wage increase.

What does corporate America say about raising the minimum wage?

A number of massive companies — Amazon and Target — under public pressure from activists, have committed to their own $15 minimum wage. Walmart has raised wages well above the $7.25 floor, but does not automatically pay all of its employees $15 per hour, although it argues that with benefits , its employees are compensated above that level.

Which lawmakers are key to this debate?

Democrats want to use something called “budget reconciliation” to pass their Covid relief bill because it allows them to maneuver around the filibuster and pass the measure with a simple majority instead of a 60-vote supermajority.

The Senate parliamentarian determined that the minimum wage hike violates the rules and can’t stay in the final package — which some Democrats said might make it easier to get it passed.

Which Democrats oppose raising the minimum wage?

Democratic Sen. Kyrsten Sinema of Arizona has said she opposed the wage hike because in her view it is not related to the pandemic.

“What’s important is whether or not it’s directly related to short-term Covid relief. And if it’s not, then I am not going to support it in this legislation,” Sinema told Politico . “The minimum wage provision is not appropriate for the reconciliation process. It is not a budget item. And it shouldn’t be in there.”

Interestingly, Arizona has already elevated its minimum wage to among the highest statewide minimum wages in the country, at more than $12. Like Democrats’ national proposal, it is attached to cost of living and rises with inflation

    This story has been updated to reflect that the minimum wage increase will not be included in the Senate version of the stimulus.

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    Living on Minimum Wage

    A national debate is underway in the United States over whether to raise the minimum wage to $15 an hour from the current $7.25 an hour today, as more than 30 million U.S. citizens now live below the poverty line.
    The last time the U.S. increased the hourly federal minimum wage was in 2009.
    According to economists and advocates, the $7 an hour rate in at least 20 U.S. states is to blame for the increase in poverty in the nation. However, in some states it’s legal to pay even lower than the minimum wage. For example, in the state of Wyoming, some residents are paid as little as $5.15 an hour or even less. And they haven’t seen a wage increase in 20 years.
    Low wages present social and economic barriers for many families in Wyoming where almost 20% of the state’s residents live in poverty. Many of them are single, lowwage working mothers, some holding down two or three jobs, to buy food and clothing for their children.
    Mike Kirsch takes us to one of the hardestworking U.S. states when it comes to earning low wages… Wyoming. 

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    Living on Minimum Wage

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