All Calculus 3 Resources
Example Questions
Example Question #1 : Absolute Minimums And Maximums
Find the absolute minimums and maximums of on the disk of radius
,
.
Absolute Minimum:
Absolute Maximum:
Absolute Minimum:
Absolute Maximum: ,
Absolute Minimum:
Absolute Maximum:
Absolute Minimum:
Absolute Maximum:
Absolute Minimum:
Absolute Maximum: ,
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
.
Certified Tutor
Certified Tutor
All Calculus 3 Resources
![Learning Tools by Varsity Tutors](https://vt-vtwa-app-assets.varsitytutors.com/assets/problems/og_image_practice_problems-9cd7cd1b01009043c4576617bc620d0d5f9d58294f59b6d6556fd8365f7440cf.jpg)