Calculus 3 : Applications of Partial Derivatives

Example Questions

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Example Question #131 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

and  are relative minima

is a relative minima 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

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

At ,

At ,

Example Question #132 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

is a relative minima

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 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 ,

Example Question #133 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

is a relative maximum

is a relative maximum

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 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 ,