Calculus 3 : Absolute Minimums and Maximums

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #1 : Absolute Minimums And Maximums

Find the absolute minimums and maximums of  on the disk of radius .

Possible Answers:

Absolute Minimum: 

Absolute Maximum: 

Absolute Minimum: 

Absolute Maximum: 

Absolute Minimum: 

Absolute Maximum: 

Absolute Minimum: 

Absolute Maximum:

Absolute Minimum: 

Absolute Maximum:  

Correct answer:

Absolute Minimum: 

Absolute Maximum: 

Explanation:

The first thing we need to do is find the partial derivative in respect to , and .

 

We need to find the critical points, so we set each of the partials equal to .

We only have one critical point at , now we need to find the function value in order to see if it is inside or outside the disk.

This is within our disk.

 

We now need to take a look at the boundary, . We can solve for , and plug it into .

We will need to find the absolute extrema of this function on the range . We need to find the critical points of this function.

The function value at the critical points and end points are:

Now we need to figure out the values of  these correspond to.

Now lets summarize our results as follows:

 

From this we can conclude that there is an absolute minimum at , and two absolute maximums at  and .

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