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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 , .
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 .
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