# Calculus 3 : Relative Minimums and Maximums

## Example Questions

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### Example Question #1 : Relative Minimums And Maximums

Find and classify all the critical points for .

Relative Minimum

Relative Maximum

Relative Minimum

Relative Minimum

Relative Minimum

Relative Minimum

Relative Maximum

Relative Minimum

Relative Minimum

Explanation:

First thing we need to do is take partial derivatives.

Now we want to find critical points, we do this by setting the partial derivative in respect to x equal to zero.

Now we want to plug in these values into the partial derivative in respect to y and set it equal to zero.

Lets summarize the critical points:

If

If

Now we need to classify these points, we do this by creating a general formula  .

, where , is a critical point.

If  and , then there is a relative minimum at

If  and , then there is a relative maximum at

If , there is a saddle point at

If  then the point  may be a relative minimum, relative maximum or a saddle point.

Now we plug in the critical values into .

Since  and  is a relative minimum.

Since ,  is a saddle point.

### Example Question #1 : Relative Minimums And Maximums

Find the relative maxima and minima of .

is a relative minimum.

is a relative minimum.

is a relative maximum.

is a relative maximum.

is a relative minimum.

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

There is only one critical point and it is at .  We need to determine if this critical point is a maximum or minimum using  and .

Since   and ,   is a relative minimum.

### Example Question #1 : Relative Minimums And Maximums

Find the relative maxima and minima of .

is a relative maximum,  is a relative minimum

is a saddle point,  is a relative minimum

and  are relative minima

and  are relative maxima

is a saddle point,  is a relative minimum

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

There are two possible values of  and .

We find the corresponding values of  using  (found by rearranging the first derivative)

There are critical points at  and.  We need to determine if the critical points are maximums or minimums using  and .

At ,

Since ,  is a saddle point.

At ,

Since  and , is a relative minimum.

### Example Question #4 : Relative Minimums And Maximums

Find the relative maxima and minima of .

and  are relative minima,   is a relative maximum.

and  are relative maxima.

and  are relative minima.

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

Squaring both sides of the equation gives us

Multiplying both sides of the equation by  gives us

There are three possible values of  and .

We find the corresponding values of  using  (found by rearranging the first derivative)

There are critical points at  and.  We need to determine if the critical points are maximums or minimums using  and .

At ,

At ,

At ,

### Example Question #1 : Relative Minimums And Maximums

Find the relative maxima and minima of .

is a relative minimum.

is a relative maximum.

and  are relative minima.

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

There is only one real value of

We find the corresponding value of  using  (found by rearranging the first derivative)

There is a critical point at .  We need to determine if the critical point is a maximum or minimum using  and .

At ,

### Example Question #6 : Relative Minimums And Maximums

Find the relative maxima and minima of .

is a relative minima,  is a relative maxima

and  are relative maxima

is a relative maxima,  is a relative minima

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

The real values of  are  and

We find the corresponding value of  using  (found by rearranging the first derivative)

There are critical points at  and .  We need to determine if the critical points are maxima or minima using  and .

At ,

At ,

### Example Question #1 : Relative Minimums And Maximums

Find the relative maxima and minima of .

and  are relative maxima.

is a relative minima,  and  are relative maxima.

is a saddle point,  and  are relative minima.

is a saddle point,  and  are relative minima.

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

Setting each factor in the expression equal to  gives us

and

The real values of  are  and

We find the corresponding value of  using  (found by rearranging the first derivative)

There are critical points at ,  and .  We need to determine if the critical points are maxima or minima using  and .

At ,

At ,

Since  and  is a minimum.

At ,

Since  and  is a minimum.

### Example Question #8 : Relative Minimums And Maximums

Find the relative maxima and minima of .

and  is a relative maxima

is a relative maxima

and  is a relative minima

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

Using a TI-83 or other software to find the root, we find that

We find the corresponding value of  using  (found by rearranging the first derivative)

There is a critical points at .  We need to determine if the critical point is a maximum or minimum using  and .

At ,

### Example Question #9 : Relative Minimums And Maximums

Find the relative maxima and minima of .

and are relative minima

,  and  are relative maxima

,and  are relative maxima

and  are relative minima

,and  are relative minima

and are relative maxima

,and  are relative maxima

and are relative minima

,  and  are relative maxima

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

Our derivatives equal  when  and .  Every linear combination of these points is a critical point.  The critical points are

We need to determine if the critical point is a maximum or minimum using  and .

minimum

minimum

maximum

maximum

maximum

maximum

minimum

minimum

### Example Question #10 : Relative Minimums And Maximums

Find the relative maxima and minima of .

is a relative minimum

is a relative maximum

is a relative maximum

is a relative minimum

is a relative minimum

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

The first order partial derivatives are

The second order partial derivatives are

To find the critical points, we will set the first derivatives equal to

There is a critical point at .  We need to determine if the critical point is a maximum or minimum using  and .

At ,

Since  and , then there is a relative minimum at .

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