### All Calculus 3 Resources

## Example Questions

### Example Question #1 : Applications Of Partial Derivatives

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: ,

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 .