Hi there.

A significant multivariate functions In practice we have seen the Taylor series.

In this section, a second basic application We'll see.

These critical points and local end The issue of value.

So ends the maximum value of the mimimum We were referring to.

Local also near a point.

Absolute extreme values is slightly different.

In this first part of her course here We'll see.

The scope of further applications We'll see.

Now with the definition of critical points Let's start.

You'll know in a univariate function You will recall the derivative is zero

that is tangential to the horizontal income counterparts.

The two-variable functions generalization

a horizontal direction is a horizontal tangent means that the tangent plane.

Now we have seen many times.

The perpendicular vector of the tangent plane.

Now may be a tangent to the horizontal plane perpendicular to

vector in the vertical plane perpendicular should be.

The k vector perpendicular direction.

So you only need to have one.

These two terms should be zero.

Accordingly, the first-order derivatives of partial derivatives

We call the critical point is zero point.

This is just one variable functions f prime equivalent to zero.

There is only one variable exists with respect to x There are two variables here with respect to x.

These two conditions to us, two unknowns

x is zero and y is zero point location will give.

Watching the smallest value it so much at a minimum that most of us

value or the maximum value definitions only In multivariate function if enough

If we know that the degree of derivative f prime is zero at the point where

Thank you a minimum value at a great were worth.

But this definition is not actually a theorem arising as a result of something.

Now we say the following small value.

An x is zero, y is zero at the point of function of x and y in the value of nearby

is smaller than the value calculated at this point We say little value.

This has brought a completely simple logic definition.

From the near point.

Around here, my blood, definition around important The concept is very important

away more than x o y is zero zero We'll find around the area.

Now the greatest value to the concept of When we arrived at

x is zero y symmetrical its exact definition function at zero

value in all points around will be greater than the value.

We also work outlier my ekstremo is called also called extreme values.

This is the minimum value or the minimum and

The maximum value is used for a joint pcs.

We will see the smallest value and the largest To find the value of the

A common process occurs spontaneously naturally.

Therefore, a significant outlier means thing.

Now local and absolute extreme values Let's open concept a bit.

In just the definition of a local extreme values x is zero

We say near zero at year-end values largest or smallest value.

They call the local end point percent.

However, if the absolute value of a closed-end

the largest or the smallest in the region values.

This single variable functions the same in In the definition of functions of several variables

of course, the same definition is operably but there are differences, but the concept is the same.

This concept of one variable in a function Let us emphasize.

As we get a function.

The function and minimum values of b and c occurs in extreme values.

As you can see from this point if you receive vicinity is this

the value of the function around the point b largest.

Similarly, as again a vicinity if you receive this c

Near the point in point c the smallest value.

They call the local extreme values and its

something for the region's borders we do not need.

But they do not necessarily one for the extreme values

closed regions that need We need boundaries.

As you can see here in a point value

from local extreme value at the point where the c smaller.

So absolute minimum value occurs in a not c.

Similarly, this gene region in D on the border

see value here, minus two to five We chose a closed zone between.

This is the absolute maximum of the function point becomes great value.

Because greater than local b.

Here are a theorem as follows monovalent function is available on.

You in a local extreme value is an absolute value Thank occurs at the boundary occurs.

This is very valuable, multivariate the same concept of functions.

And of course, here it is more difficult to draw There are functional differences.

Already we mostly new

requires knowledge of the local extreme values We are interested in the type.

But it's local and absolute extreme values

useful to distinguish between the concepts of a I've seen at this stage.

Now let's start from the smallest value.

As a univariate function thought and

How we found the conditions there, the Let us remember.

x is zero the value of the function call x is smaller than near.

Where f x is zero on the right side

If we now see f x f x minus zero will be greater than zero.

So f x f x is zero minus zero large.

Near call.

You mean I can open a Taylor series around and

a good close approximation would give we think.

Let's open it the first few terms.

The first term f x will be zero, of course.

It also will take f x is zero.

Subsequent first-order derivative terms.

Subsequent second order derivative terms and so it goes on.

So f x f x is zero minus x

be valued at zero plus the be obtained by examining the terms.

Now here we are watching this right now.

x x is zero can be left to the right in may be on a line.

So x if x is greater than that plus would be valuable.

If a small minus would be valuable.

Therefore, both the x minus x is zero

can be a plus and minus all may be for points.

Therefore, keep this sign only species not possible.

If f is base plus the value of x x For values less than zero, this

would have a negative value due to the notion of residual ensure not possible.

The only cure for that prime 0 In the case requires one variable.

In multivariate functions from the two

Starting the same logic will conduct is here.

Following the call for qualification.

Since we dropped the first term with the term wonder Can we provide added value?

Now, right here, as we see here squared

There hence the term x minus x is zero

plus though, albeit minus the square so We're here to give an added value.

Therefore, if the second order derivatives plus

plus it is valuable to be valuable we provide.

This is enough reason?

Because call around.

x minus x is zero small.

x minus x is zero because it is smaller first

Before we look at the first-order terms We're looking at the first force.

X squared minus x is zero because it is less even smaller.

Therefore, this is the highest at the general behavior becomes evident with large customers.

But again this term after the fall of the We're looking for key terms.

Because then the term cubeful will be.

If x is less than minus x is zero terms cubeful will be smaller than plaid.

If you assume that this is a zero point.

So it will be a square.

Cubes in a thousand will be.

Thus, the term general behavior

With this term therefore will be determined.

Single variable such a logic more There are complicated ways, but it

including a requirement that the main idea

the sufficiency condition is a condition we find.

If we're looking for if we x the maximum at zero

value that is greater than minus sign will be.

Less well again to ensure that requirements will not change the condition.

Because it can be plus minus the I might.

That the only remedy is zero.

When we look to these terms plus to be checked for

The only condition means that the second derivative of the term is negative.

This vision comes our little awry.

I come at least.

Maybe for you, do you think it is.

When we say the biggest plus it Valuable

We think like you should be, whereas minus happening.

When we say it minus the smallest As should be valuable düşünüyosun no

In the first reaction of people but vice versa We see that in mind this simple.

Now that the two functional equations

We'll proving the two different pathways: We'll proving.

Important because each one will teach application

also have the skills, mindset, too.

How to value the smallest single variable

here in the first derivative of the function is zero but has two first derivatives.

Both with respect to x and y according to.

Two of them are supposed to be zero.

The second condition here as soon We will see that

Still a further plus a term that The term is supposed to be a plus.

See this second derivative of univariate derivatives was supposed to be a plus.

Here again, plus the smallest value for is going on.

But even this second condition as well as is there.

The first derivative is zero in the univariate from olmasıda were finding the location of the x is zero.

Here, too, this time two unknown is there.

x is zero and y is zero.

But here are two equations

From these two equations we therefore x zero

and y is zero here our two known

We can easily produce at least conceptually.

Some accounts may be too complicated course but it's not much importance.

Important to both of these two unknown to understand that equation.

Most of it is great value for

but in the opposite occurs adequacy requirements still the same.

The first order derivatives in one variable SFr was happening.

Here again, the first order derivatives zero and

As you can see this is a critical point indicate that.

Shows that the horizontal tangent plane In both cases.

Sufficient condition at the same univariate

minus sign as the second derivative is going on.

But another condition in addition to more is there.

They are also very systematic why you came

in a form easily We'll see.

Now here will do one of two ways to prove I said.

One is that the Taylor expansion applications

supremely important because a point to understand the vicinity

Are you working in and around the first few

terms of the nature surrounding us immediately that gives.

The second derivative of the chain saw.

As an important method of calculation.

This is a good practice both these

it reinforces the concept of chain derivatives and

to a single variable functions doing by reducing

One thing we know that by reducing is doing.

For that reason, a different logic, but both corroborate each other.

Naturally we come to the same conclusion.

Now we are doing now with Taylor series:f x to y, let's open the series.

First term value of x in y is zero zero will be.

Drops.

Then the first order terms then

second-order terms, then on forever This continues.

The same logic in one variable, we just plus we want to be valued.

To be valuable plus x minus x zero plus and

would be minus y minus y is zero for both pros and cons may

For the x's and y's also their is independently a single

remedy this in order to keep it in surplus The term is not zero.

This is again emerged as a necessary condition involved.

Now the second, so this terms

thought to look at the second order terms need.

We need to look to the second moment x the former is smaller x is zero

squared terms of the terms of this cube more dominant factor

terms this to be a sign of general will determine behavior.

You'll recall from the Taylor series x Because squared terms

y, and x and y also have squared terms there are terms that are multiplied.

They sounded two times.

Because both f x by y so ago after x'y

derivative with respect to y or y x li-bent as you terms.

E them as equal to each other

It does not matter if the continuity conditions if provided.

CLEAR he initially saw theorem it

this explains these two coefficients.

Now here's a nice application We will see to the square of completing a method.

I hope everyone looked at some preliminary information.

Orr is one of the fundamental techniques

completion method to this frame Provided.

We also see how to do it in here In our session together.

Or If y'all If y'all but I forgot to look I guess but I'm sure

Those who have looked at everyone but forgot to look I might.

Seeing a beautiful application We will remember.

Now let's say it is.

A value of one-half of the course mark, not contribute.

This quadratic polynomial and other mixed

To be squared terms plus We'll search terms.

As in the method of completing the square to 're doing.

I wrote here again.

Let's get out of here, for example, f x x.

f x where x is the number of times when you get out will be one in a bracket.

Here of course we need to divide the f x y x x the.

The term also need to partition.

Now we are doing the following into squares to complete.

See here that the start of a frame The first two terms.

we take the square of the first term of a plus b is will be square.

Plus going to be multiplied twice.

Therefore we x where x is the old zero-plus y is zero minus half times that of the year

and if we consider that if we take the square this term will produce the first frame.

See twice the product of these two x minus x

this term will be reset to zero y minus y will produce.

Two factors will be.

The coefficients of the same this term will produce.

But will that suit the more because an extra term It was a square.

Both times it was a b.

BI will also b squared.

We produce this extra term back we need to take.

To undo an interesting coincidence Or a structural phenomenon emerges.

See here also have zero years old this year squared.

Here, too, when we received the frame y minus y The frame will be zero.

Therefore, a common term here occurs.

But here the same as in the first term be it in terms we can remove the square.

See, here we see the square.

There was a very beautiful structure.

The following respects:one in brackets There are full-frame terms.

This will be valuable is always a plus.

When we look to y minus y is zero plus valuable all the time.

Therefore, the positive value of this to provide

plus valuable for the initial term will be.

For it plus the value of a to provide

plus the value of this coefficient is only will be sufficient.

Because here there is already a square.

Surplus value.

There are a total surplus value already here.

Therefore, the main sign of it here will occur.

Plus valuable under any circumstances be the.

So it's positive valence We need to yababilme.

I am writing this here.

See the following occurs here.

There are a partnership in the denominator, but not full.

Here are the square of f x x.

Here are just their own.

Therefore, the common denominator of the square of f x x Get a f x x multiplied by coming here.

As you can see this term.

Minus the square of the mixed derivative term future.

F x is x squared in the denominator for The effect will not be his mark.

It can reduce.

Therefore it is reduced, plus valuable be

in the square brackets small.

This means that the surplus be valuable an essential condition.

It is also here that BI brace plus If the total value

plus it to be valuable to f x x'ind plus the value should be.

If we say these conditions have obtained the following

we are:The first order derivatives to zero will be.

The same requirements as in the one variable condition

We have two unknown x is zero and y is zero.

These critical point, end point coordinates.

Two of them to determine We've got the equation.

So the two unknowns from two equations In principle we can solve it.

Sometimes accounts can be mixed or something, but it is a detail that is.

If this qualification requirement, see here The term also has a good structure.

As a determinant of its We can arrange.

Y'l on the x and y of X, the first diagonal partial derivatives

If we put in on the second diagonal complex, mixed

Putting terms derived x y and y x

are equal and thus to Coming frame anyway.

See, we get it.

If you open these determinants of the first diagonal on

The first term gives the product of these terms.

Less of those over the second diagonal multiplying it

for the same term is squared it giving.

Therefore, the term more in mind moreover easy to keep

mathematics in terms of a more regular We see here as a term.

Plus, when this term means that f x

When the minimum value of x plus is reached.

We are at the same operations from the beginning large

value minus sign here if we had to would be found.

To ensure that the minus sign brackets

nevertheless necessarily a plus sign in should be from the following aspects.

There is a square where it can not do anything it always will be in surplus.

You can do this in a negative value, perhaps, but all y and y,

Reset all this for years minus y minus You can not keep valuable.

For that reason, plus be valuable here have.

But there is another factor we used.

This is the second derivative with respect to x.

Therefore, this should be less valuable.

It occurs when the maximum value that the end values occurs.

A second type that is the largest and most except for a small

minus sign of the determinant of a kind be.

This is the determinant minus sign for x of x neighborhood

points do not mean much for the value because the negative

If checked, you can select the x and x is zero where you can bring added value.

Or you can choose so, where negative You can return mark.

Therefore with this greater işaretlig You do not have the opportunity to play.

Only now this term is checked for negative x

No sign of x to determine the meaning terms.

Therefore, a kind of a third critical points involved.

This is called a saddle point.

We will see this in more detail.

He can revive a saddle point Consider the saddle on the horse.

Upwardly along the spine at a There is a curve like a parabola.

When we look at the side downwards There outgoing curves.

We initially used the first

We have seen in our department a certain surface a hill

that's like a valley or pit saddle comes a point opposite the valley.

Although two peaks between the two peaks Think of the valley.

Remove the two sides of this valley you up As you go down in the saddle.

Just another example of our culture a slender

tea cup final and we call away Lay it horizontally.

As you can see in one direction, upwards While going down in the other direction.

He's never at a critical point in the largest You can not create or smallest point.

What is the biggest for him here, what the is small.

For this reason, it is another kind.

It was called a saddle point.

Now you have a second way to prove I said.

I'm here to do şşle of chain derived.

Here on this by selecting a curve function of one variable are demoting.

By reducing the total derivative univariate

on the concept of gene to the same results We will achieve.

Gene completion method to square be used.

But leave it to the next session I want.

We will keep in remembrance of these two variables

function of the condition of the two requirements Coming equation.

Two conditions for qualifying condition supposed to provide.