### All Calculus 3 Resources

## Example Questions

### Example Question #131 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

**Possible Answers:**

and are relative minima

and are saddle points

is a relative minima and is a relative maxima

is a relative maxima and is a relative minima

**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

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

At ,

Since , is a saddle point.

At ,

Since , is a saddle point.

### Example Question #132 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

**Possible Answers:**

is a relative minima

is a saddle point

is a relative minima

is a saddle point

**Correct answer:**

is a saddle point

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 exponential part of each expression cannot equal , so each derivative is only when and . That is and .

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

At ,

Since , is a saddle point.

### Example Question #133 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

**Possible Answers:**

is a saddle point

is a relative maximum

is a relative maximum

is a saddle point

**Correct answer:**

is a saddle point

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 exponential part of each expression cannot equal , so each derivative is only when and . That is and .

The critical point is . We need to determine if the critical points are maxima or minima using and .

At ,

Since , is a saddle point.

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