AP Calculus AB Maximum & Minimum Values of a Function

Hello, and welcome back to www.educator.com, welcome back to AP Calculus.0000

Today, we are going to talk about the maximum and minimum values of a function.0005

A function on a given domain, it is going to achieve different types of max and min.0009

We are going to talk about an absolute max, an absolute min, local max, and local min.0015

Let us jump right on in.0020

Let us start off by just looking at this graph.0024

Let me go ahead and tell you what this graph is.0026

It is what the function is actually, that this graph represents.0030

Let us go to black here.0041

Let us call it f(x).0043

Here f(x) is equal to 3x⁴ - 14x³ + 15x².0044

The domain is restricted on this one.0056

The domain happens to be -0.5.0061

X runs from -0.5 and it is less than or equal to 3.5.0065

We know that a function is complete, when you actually specify its domain.0073

Generally, for the most part, we do not talk about domains but domains really are important.0078

In this case, even though this function is defined over the entire real line, we are going to restrict its domain here.0082

Notice that this is less than or equal to.0090

The endpoints do matter, so it is defined.0092

We have a point there and a point there.0096

Like I said, we are going to be talking about an absolute max, an absolute min, a local max, and local min.0100

I’m going to go through a few of these graphs and talk about it informally.0106

And then, I will go ahead and give formal definitions for what they are.0110

The formal definitions are there just for you, they are in your book.0113

Our discussion right now in the first couple of minutes is going to make clear exactly what these things are.0117

The absolute max of a function on a given domain is exactly what it sounds like,0123

it is the highest value that f(x) takes on that domain.0126

In this particular one, this point right here, which happens to be 3.5 and 33.69.0131

This is the absolute max.0141

On the domain, it is the highest value that f(x) actually achieves.0147

We know that if did not restrict the domain, it would go up into infinity.0151

In that case, there is no absolute max, there is no upper limit that we can say.0157

But here, we can because we have restricted the domain.0161

Let us talk about the absolute minimum.0165

The absolute minimum, the lowest value that a function actually takes on its domain happens to be over here.0168

This point is 2.5 and -7.81.0174

This is an absolute min, it is the lowest value that it takes on its domain.0181

Let us talk about something called a local max and local min.0189

A local max and a local min, that is where the function achieves a local max and local min at a point x in the domain.0192

Such that, if you take some little interval around that particular x, that f(x) is bigger than every other number.0202

Or the f(x) is smaller than every other number.0213

In this case, this right here, this point which happens to be the point 1,4, this is a local max.0216

It is a local max because if I move away from the point 1, if I move a little bit this way or a little bit this way,0228

notice the function is lower than this, the function is lower than this.0235

For a nice small little region around the point, if at that point the function achieves the maximum value that it can,0240

in that little bit, it is a maximum locally speaking.0248

Locally meaning some little neighborhood around that point.0252

Here this is an absolute max, it is also called the global max.0255

Absolute min also called the global min.0259

Overall, what is the biggest?0261

Locally, that is this.0263

This is a local max.0265

Interestingly enough, this point right here also happens to be a local min.0267

This point 2.5, if I take a little neighborhood around 2.5,0273

within that little neighborhood locally around the 2.5, this is the lowest point.0279

Because if I move to the right, the function is bigger.0284

If I move to the left, the y value of the function is bigger.0288

This also happens to be a local min.0291

That can happen, a local min can be an absolute min.0294

A local max can be an absolute max.0298

You can have more than one local min and local max.0301

You can have only one absolute max and absolute min.0305

You might have no absolute max, no absolute min, but you have a bunch of local max and min.0308

You might have no local max and min but you might have an absolute max and absolute min.0314

These all kind of combinations.0318

In this particular case, we have an absolute maximum that it achieves.0321

We have an absolute minimum, also happens to be local minimum.0327

We have a local maximum.0332

This end point over here, it does not really matter.0333

It is the y value is someplace in between.0336

That is it, that is what is going on with absolute max, absolutely min, local max, and local min.0338

We also speak about the point at which the function achieves its absolute max, local max.0346

In this case, this function achieves an absolute max at x = 3.5.0352

It achieves an absolute min at a local min at x = 2.5.0357

It achieves a local max at 1.0361

The values themselves are the y values, 33.694 and -7.81.0364

Let us look at another function here.0372

Once again, a graph can have all or none of these things.0379

Let me go ahead and write that.0381

I think I will use blue.0384

A graph can have all, some, or none of absolute max, absolute min, and the local max and min.0390

In this particular case, let me see, what function I have got here, it looks like the x² function.0415

Here we are looking at the function y = x².0422

Once again, we have restricted its domain.0426

2x being greater than or equal to 0.0429

This point is absolutely included and this just goes off to positive infinity.0433

In this particular case, is there a highest point on this domain?0439

No, because this goes up into infinity, we cannot say that there is an absolute max.0444

There is no absolute max.0449

Is there an absolute min?0454

Yes, there is, this is the lowest point overall on this domain.0455

It is lowest y value.0460

This point 0,0 is the absolute min.0462

Are there local max or min?0467

No, there are not.0470

There is no local max, there is no local min.0472

You might think yourself, could this not be considered a local minimum?0478

No, a local minimum requires that at a point, in this particular case 0,0,0482

that there is actually an interval to the left and to the right of it, which satisfies the conditions.0487

In other words, yes, if I move to the right, it is true.0494

The function gets higher in value but it is not defined to the left.0497

It is not defined to the left but for a local, I need something that is defined around.0501

An interval around it has to surround that thing.0507

The local min always looks like a little valley.0511

A local max always looks like a crest of the hill.0514

That is it, that is local max and local min.0518

In this case, absolute min and no absolute max, no local max, no local min.0522

Let us take a look at another function here.0528

This particular function right here, we have not restricted the domain at all.0541

This goes off to positive infinity, this goes down to negative infinity.0545

It looks like some sort of a cubic function.0549

I actually did not write down what function this is but that is not a problem.0552

We are here to identify graphically absolute max and min, and local max and min.0555

In this particular case, there is no absolute max and there is no absolute min.0560

However, we do have a local min and we do have a local max.0572

Yes, there is a local minimum and it looks like it achieves that minimum at x = 2.0577

There is a local maximum and it looks like it achieves that maximum at x = -2.0583

The maximum values and the minimum values happen to be the y values.0591

Whatever that happens to be, it looks like somewhere around 16, something like that.0596

The same thing around here.0601

That is it, local min, local max, no absolute max, no absolute min.0603

Let us go ahead and give some formal definitions to these concepts, because you are often going to see the formal definitions.0616

Mathematics is about symbolism.0621

We have talked about these things informally, geometrically, let us give them some algebraic identity.0623

Formal definitions, we will let f(x) be a function.0650

We will let d be its domain.0672

Let us define what we mean by absolute max.0680

The absolute max also called the global max.0685

If there is a number c that is in the domain such that0692

the value of f at c is actually bigger than or equal to the value of f(x) for every single x in the domain.0710

Then, f achieves its absolute max at c.0726

The value f(c) is the absolute maximum value.0741

Once again, if there is a number c that has to be in the domain,0753

such that f(c) is bigger than f(x) for all the other x in the domain, then f achieves its absolute max at c and f(c) is the absolute max.0757

That is it, it is the largest y value that the function takes in the domain.0769

This just happens to be the formal definition.0774

Let us give a definition for absolute min which is also called a global min.0778

You can imagine, it is going to be exact same thing except this inequality is going to be reversed.0784

Global min, if there is … everything else is the same.0791

Such that floats that f(c) is actually less than or equal to f(x), for all x in d.0802

Then, f achieves its absolute min at c and f(c) is that absolute min.0824

Nothing strange, completely intuitive.0847

You know what is going.0849

But again, it is very important.0850

To start the formal definitions in mathematics, very important because you want to be very precise0853

about what is it that we are talking about.0860

We want to be able to take intuitive notions and put them into some symbolic form.0861

Let us go ahead and define what we mean by local max and local min.0868

You know what, I think I’m going to go all these wonderful colors to choose from.0872

I’m going to go to black for this one.0877

Local max also called the relative max.0883

The definition is, if there is a c in the domain such that f(c) is greater than or equal to f(x),0893

for all x not in d, for all x in some open interval around c.0914

Remember what we said, once you have a point c, we have to take some interval around this point c.0927

If I go to the right of c and to the left of c, that the function drops, that is that.0949

Now it is not over everything, it is just locally speaking, a little bit.0955

If there is a c and d such that f and c is greater than or equal f(x) for all x in some open interval around c,0960

then f achieves a local max at c and f(c) is that local max.0970

The definition for local min which is also called a relative min.0989

Everything is exactly the same.0998

I’m just going to do if … the defining condition is f(c) is less than or equal to f(x) for all x in some open interval around c.0999

Then, f achieves its local min at c and f(c) is that local min.1013

Nothing strange, let me go back to blue.1038

The absolute max value and the absolute min values are also called the extreme values.1045

We are going to list a important theorem called the extreme value theorem.1075

We have the extreme value theorem.1090

If f is continuous on a closed interval ab,1106

then f achieves both an absolute max and an absolute min on ab.1126

Very important, the two hypotheses of this theorem that, if f is continuous,1152

f has to be continuous and it has to be a closed interval.1158

If those two hypotheses are satisfied, then the conclusion is that f has an absolute max and an absolute min on that closed interval.1163

It might be inside the interval, in other words it might be a local max or local min were achieved its highest.1176

Or it might actually be at the endpoints because the closed interval, the endpoint are part of the domain.1182

If it is continuous, if it is closed interval, then both absolute max and absolute min are achieved.1188

If one or both of the hypotheses are not satisfied, you cannot conclude that it has a max or a min.1195

May or may not, but you cannot conclude it.1201

If a function is not continuous on its domain, if the interval is not closed, all bets are off.1204

Let us go ahead and take a look at a couple of examples of that.1211

I will do this in red.1214

We have got something like that.1216

Let us go something like that.1221

Here is a and here is d, this is a closed interval.1230

The function is continuous.1234

Therefore, it achieves its absolute max and absolute min somewhere on this interval, based on the extreme value theorem.1236

In this particular case, here is your absolute min,1245

I’m reversing everything today.1249

This is your absolute max and this is your absolute min.1251

In this particular case for this function, they also happen to be local max and local min but that does not matter.1259

Continuous function, closed interval, it achieves an absolute max and it achieves an absolute min.1265

Let us look at another graph.1272

This is a, this is b, continuous function, closed interval.1280

We have the absolute min, we have the absolute max, on that closed interval always.1286

Let me draw a little circle, something like that.1311

This is a, this is b, this is a closed interval.1317

However, the function is not continuous.1322

Therefore, it does not achieve both an absolute max and an absolute min.1325

Here I see some absolute max but there is no absolute min.1332

Because in this particular case, this is an open circle.1336

It gets smaller and smaller but we do not know how small it actually gets.1340

If there is no absolute smallest value of y on this.1348

There is not because it is discontinuous there.1353

It does not satisfy the hypotheses, so it does not apply.1356

We will do one more.1366

We will take the function f(x) = 1/x, your standard hyperbola.1370

Here there is no max, there is no absolute max, and there is no absolute min.1379

The reason is it is continuous but there is no closed interval.1385

I have not specified a closed interval that has well defined endpoints.1391

This is going to keep climbing and climbing.1397

This is going to keep dropping and dropping.1399

You might think yourself, wait a minute, in this particular case, cannot I just say that 0 is an absolute min?1403

No, what 0 is a lower bound.1408

In other words, the function will never drop below 0.1414

But I cannot say that there is a smallest number that is still bigger than 0, that this function will hit.1418

It is going to keep getting smaller and smaller, heading towards 0.1425

In this case, like that one, this number is a lower bound on this function.1429

In other words, it will never be lower than that.1434

But that does not mean that that is in absolute minimum because it does not achieve its minimum.1437

Because for every number I find that is small, that is close to this lower bound, like close to 0 over here,1442

I can find another number smaller than that closer to 0.1448

That is the whole idea of this infinite process.1452

There is a very big difference.1455

A lower bound or upper bound is not the same as absolute max and absolute min.1457

Absolute max and absolute min, they have to belong to the domain.1461

The x values at which the absolute max and absolute min are achieved, they have to be part of the domain.1468

Let us move on, one more theorem here.1480

Let us go ahead and leave it in red.1483

If f(x) has a local max or a local min at c in the domain of the function, then, f’ at c is equal to 0.1488

All that means is the following.1522

We already know what local max and local min look like.1525

Local min is a valley, local max is a crest.1527

If I have some function like this, this is a local max and this is a local min.1531

We will call this c1, we will call this c2, whatever the x value happens to be.1539

This says that at the local max and at the local min, the slope is 0.1544

The derivative f’ at c is 0, the derivative is 0, the slope is 0.1551

We can see it geometrically.1557

We are going to have a positive slope, from your perspective, if you are moving from negative to positive.1558

Positive slope, it is going to hit 0 and it is going to go down like this.1563

That tells us that we have a crest, a local max.1567

Then this one, local min.1571

That is it, that is all this theorem says.1573

Let us go ahead and give the definition.1577

Definition, something called a critical number or a critical value.1583

The number c that is in the domain such that, f’ at = 0 or f’ at c does not exist.1605

A critical number or a critical value of the function, it is a number in the domain such that it is a number c in the domain,1631

such that f’ at that number is either equal to 0 or f’ at c does not exist.1636

In this particular case, these values, f’ of these values is definitely 0, it is a horizontal slope.1645

These are critical values.1652

An example of one where it does not exist is the absolute value function.1656

Absolute value function goes that way and it goes that way.1660

It is continuous there at 0 but is not differentiable there.1667

Because it is not differentiable there, that is a critical value of the absolute value function.1675

F’ at c = 0 or f’ at c does not exist.1683

If it is not defined there, that is not considered a critical value.1687

It has to actually be defined there.1690

It has to be in the domain, that is important.1692

If it is not part of the domain, then all bets are off.1697

Let me write this a little bit better.1710

Let us do it in red.1713

The procedure for finding the critical values, very simple.1718

The procedure for finding the critical values of a function f(x).1723

Find the derivative f’(x) into = 0.1735

Set f’(x) equal to 0 and solve for all values of x that satisfy this equation.1751

This equation, the only other thing that you have to watch out for is place on the domain1771

with the function is not differentiable.1778

Other than that, find the derivative, set the derivative equal to 0, and you are done.1782

Let us go ahead and actually do an example of this.1789

I’m going to call this example 1.1792

Example 1, find the critical values of f(x) = x + sin(x).1797

We know that f(x) = x + sin(x).1820

F’(x) is equal to 1 + cos(x).1827

We take the derivative and we set it equal to 0.1833

1 + cos(x) is equal to 0 and we solve.1836

Cos(x) = -1.1841

Therefore, x = π.1846

Let us just stick to a particular domain, let us go from 0 to 2π.1852

We know that it repeats over and over again but that is fine.1857

We will stick to 0 and 2 π.1860

In this particular case, in this domain, the critical values are x = π.1862

That is a place where the derivative is equal to 0.1869

Are there any places where this function is actually not differentiable on this domain?1872

No, the cosign function is discontinuous everywhere and it is differentiable everywhere.1876

I do not have to worry about that other part of the definition of critical value.1883

I just have to worry about taking the derivative and setting it equal to 0, and solving.1887

Let us list the procedure for finding.1896

Do not worry about it, as far as this example is concerned,1902

this particular lesson is just the presentation of the material with a quick example.1905

The following lesson is going to be many examples of what it is that we are doing.1910

There are going to be plenty examples, I promise you.1916

Now we are going to talk about the procedure for finding the absolute max and absolute min.1919

Remember, we have that extreme value theorem.1924

We said that, if a function is continuous on a closed interval, that it achieves its max and min, absolute max and min.1925

How do we find that absolute max and min?1934

Here is how you do it.1935

Procedure for finding the absolute max and absolute min of f(x) on ab, the closed interval.1936

One, find the critical values of f(x), that is what we just did, that procedure.1955

Evaluate the original function, evaluate f(x) at each critical value.1970

Third step, we want to evaluate f(x), the original function, that is going to be the hardest part,1988

especially now that we get into this max and min.1995

We are going to be talking more about derivative, setting them equal to 0, using it to graph.1996

You are going to be dealing with functions, their derivatives, the first derivative and the second derivative.2002

You are going to find certain values and you are going to be plugging them back in.2007

Which do you plug it back in?2010

You do have to be careful.2012

It is an easy procedure but just make sure the values that you find, you are plugging back into the right function.2013

When we find the critical values, we are going to be using f’(x), setting it equal to 0.2020

When we find those values, we are going to be actually to be putting them back into f(x) not f’(x).2025

The third part is evaluate f(x) at each end point.2032

In other words, you want to evaluate f(a) and f(b).2045

Now you have a list of values.2057

You have a list of values f(x) at each critical value.2061

You have f(a) and you have f(b).2064

Of these tabulated values for f(x), the greatest one, the greatest is the absolute max and the smallest is the absolute min.2072

That is it, nice and simple.2099

Let us do an example.2102

Let me skip this graph.2130

Now example 2, what are the absolutely max and absolute min of f(x) is equal to x + 4 sin x on 0 to 2 π.2132

Let us go ahead and take f’.2171

F’(x) is equal to 1 + 4 × cos(x).2173

We set that equal to 0 because our first step is to find the critical values of this function.2180

Critical values, we take the derivative and we set it equal to 0.2184

I have got cos(x) is equal to 1/4 that means that x is equal to the inverse cos of 1/4 or 0.25.2189

On the interval, from 0 to 2 π, I find that x is equal to 1.82 rad or 104.5°, if you prefer degrees.2194

Or we have 4.46 rad, I’m sorry this is going to be -1/4, 255.5°.2218

These are our critical values.2233

We want to evaluate f at those critical values, that is our second step.2237

F(1.82) is equal to 5.70.2242

In other words, I put these back into the original function to evaluate it.2249

F(4.46) is equal to 0.59.2256

I’m going to evaluate at the endpoints, 0 and 2 π.2264

F(0) is equal to 0 and f(2 π) = 2 π which = 6.28.2271

I have 5.7 and 0.59, 0 and 6.28.2283

The absolute max is the biggest number among these.2289

The absolute max is equal to 6.28.2294

The absolute max happens at x = 2 π.2300

The absolute min value, that is going to be the smallest number here is 0, and that happens at x = 0.2309

That is all, find the critical values, evaluate the function of the critical values.2322

Evaluate the function at the endpoints, pick the biggest and pick the smallest.2327

You are done, let us see what this looks like.2330

Here is the function.2336

This right here is your f(x) and I decided to go ahead and draw the derivative on there too.2339

This is f’.2346

This f(x) here, this is the x + 4 sin x.2352

This f’(x), this is equal to 1 + 4 × cos(x).2359

You know it achieves its maximum value at whatever it happened to be which was 2 π, I think, and its minimum at 0.2369

There you go, that is it.2377

Over the domain, you are done.2378

2 π would put you on 6.28, somewhere around here.2383

Sure enough, that is a the highest value because we are looking at that.2394

This is going to be the lowest value, that is all.2403

We have the critical values where the derivative equal 0.2407

Those were here and here.2410

In other words, whatever that was, the 1.82 and I think the 4.46, those are local max and min.2414

Local max and local min but they are not absolute max and absolute min.2420

We have to include the endpoint.2424

In this particular case, it is the end points at which this function achieves its absolute values, its extreme values.2427

Thank you so much for joining us here at www.educator.com.2436

We will see you next time so that we can do some example problems with maximum and minimum values.2438

Take care, bye.2444