### All Calculus 3 Resources

## Example Questions

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

Find and classify all the critical points for .

**Possible Answers:**

Relative Maximum

Relative Minimum

Saddle Point

Saddle Point

Saddle Point

Saddle Point

Saddle Point

Saddle Point

Relative Minimum

Relative Minimum

Saddle Point

Saddle Point

Relative Minimum

Saddle Point

Relative Maximum

Relative Minimum

Relative Minimum

Saddle Point

Saddle Point

Saddle Point

**Correct answer:**

Relative Minimum

Saddle Point

Saddle Point

Saddle Point

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.

Since , is a saddle point

Since , is a saddle point

### Example Question #2 : Relative Minimums And Maximums

Find the relative maxima and minima of .

**Possible Answers:**

is a relative maximum.

is a relative minimum.

is a relative minimum.

is a relative maximum.

**Correct answer:**

is a relative minimum.

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 #3 : Relative Minimums And Maximums

Find the relative maxima and minima of .

**Possible Answers:**

is a saddle point, is a relative minimum

and are relative maxima

is a relative maximum, is a relative minimum

and are relative minima

**Correct answer:**

is a saddle point, is a relative minimum

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 .

**Possible Answers:**

and are relative minima, is a relative maximum.

, and are relative maxima.

, and are relative minima.

, and are saddle points.

**Correct answer:**

, and are saddle points.

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 ,

Since , is a saddle point.

At ,

Since , is a saddle point.

At ,

Since , is a saddle point.

### Example Question #5 : Relative Minimums And Maximums

Find the relative maxima and minima of .

**Possible Answers:**

is a relative minimum.

and are relative minima.

is a relative maximum.

is a saddle point.

**Correct answer:**

is a saddle point.

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 ,

Since , is a saddle point.

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

Find the relative maxima and minima of .

**Possible Answers:**

and are relative maxima

and are saddle points

is a relative minima, is a relative maxima

is a relative maxima, is a relative minima

**Correct answer:**

and are saddle points

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 ,

Since , is a saddle point.

At ,

Since , is a saddle point.

### Example Question #7 : Relative Minimums And Maximums

Find the relative maxima and minima of .

**Possible Answers:**

is a relative minima, and are relative maxima.

is a saddle point, and are relative minima.

, and are relative maxima.

is a saddle point, and are saddle points.

**Correct answer:**

is a saddle point, and are relative minima.

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 ,

Since , is a saddle point.

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 .

**Possible Answers:**

is a relative maxima

and is a relative minima

is a saddle point

and is a relative maxima

**Correct answer:**

is a saddle point

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 ,

Since , is a saddle point.

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

Find the relative maxima and minima of .

**Possible Answers:**

, , , ,, , and are relative minima

, , and are relative maxima

, , and are saddle points

, , , ,, , and are relative maxima

, , and , , , and are saddle points

, , , ,, , and are relative maxima

, , and are saddle points

, , and are relative minima

, , , ,, , and are saddle points

, , and are relative minima

, , and are relative maxima

**Correct answer:**

, , , ,, , and are saddle points

, , and are relative minima

, , and are relative maxima

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 .

, , ,

Saddle point

minimum

minimum

Saddle point

, , ,

maximum

saddle point

saddle point

maximum

, , ,

maximum

saddle point

saddle point

maximum

, , ,

saddle point

minimum

minimum

saddle point

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

Find the relative maxima and minima of .

**Possible Answers:**

is a relative minimum

is a relative maximum

is a relative minimum

is a relative maximum

**Correct answer:**

is a relative minimum

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 .