Calculus 3 : Relative Minimums and Maximums

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Relative Minimums And Maximums

Find and classify all the critical points for .

Possible Answers:

 Relative Minimum

 Relative Minimum

 Saddle Point

 Saddle Point

 Relative Maximum

 Relative Minimum

 Saddle Point

 Saddle Point

 Saddle Point

 Saddle Point 

 Saddle Point

 Saddle Point

 Relative Minimum

 Saddle Point 

 Saddle Point

 Saddle Point

 Relative Minimum

 Saddle Point 

 Relative Maximum

 Relative Minimum

Correct answer:

 Relative Minimum

 Saddle Point 

 Saddle Point

 Saddle Point

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.

 

Since  is a saddle point

Since  is a saddle point

Example Question #122 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

Possible Answers:

  is a relative minimum.

  is a relative maximum.

  is a relative maximum.

  is a relative minimum.

Correct answer:

  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 #2 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

 is a saddle point,  is a relative minimum

 and  are relative minima

 and  are relative maxima

 is a relative maximum,  is a relative minimum

Correct answer:

 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 #3 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

 and  are relative minima.

 and  are saddle points.

 and  are relative minima,   is a relative maximum.

 and  are relative maxima.

Correct answer:

 and  are saddle points.

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 ,

Since  is a saddle point.

At ,

Since  is a saddle point.

 

At ,

Since  is a saddle point.

Example Question #4 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

  is a saddle point.

  and  are relative minima.

  is a relative minimum.

 is a relative maximum.

Correct answer:

  is a saddle point.

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 ,

Since  is a saddle point.

Example Question #5 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

 is a relative minima,  is a relative maxima

 is a relative maxima,  is a relative minima

 and  are saddle points

 and  are relative maxima

Correct answer:

 and  are saddle points

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 ,

Since  is a saddle point.

 

 At ,

Since  is a saddle point.

Example Question #6 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

  is a saddle point,  and  are relative minima.

  and  are relative maxima.

 is a relative minima,  and  are relative maxima.

  is a saddle point,  and  are saddle points.

Correct answer:

  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 ,

Since  is a saddle point.

 

 At ,

Since  and  is a minimum.

 

At ,

 

Since  and  is a minimum.

Example Question #7 : Relative Minimums And Maximums

Find the relative maxima and minima of .

Possible Answers:

 is a relative maxima

 is a saddle point

 and  is a relative minima

 and  is a relative maxima

Correct answer:

 is a saddle point

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 ,

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 saddle points

 and are relative minima

 ,  and  are relative maxima


 ,and  are relative maxima

 and are saddle points

  and  are relative minima

Correct answer:


 , ,and  are saddle points

 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 .  

 

 

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 #8 : 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

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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