Find and classify the absolute extrema of the function K(z)= \sqrt{81-z^2}on the domain D=(−9,9). If the function doesn't have an absolute extremum, write None for your answer.

Question

Hello,

I am having trouble with this problem. I tried to search it up but I can not seem to find a tutorial and the example is not that clear. All the tutorials I have seen are referring to an interval and not a domain. I tried to solve it and my absolute maximum and minimum we're wrong, I got Absolute maximum=(-9,0),(9,0) and Absolute minimum=(0,-9).

The example problem I got was Find and classify the absolute extrema of the function K(z)= \sqrt{900-z^2}on the domain D=(−30,30). If the function doesn't have an absolute extremum, write None for your answer.

Also if the function appearance is confusing, it is a square root of 81-z^2.

Thank you so much to whoever is answering.

Answer 1

FYI- A domain and an interval in this context is the same thing.

Step 1: Take the derivative of the function and set it equal to zero.

K = sqrt(81-z²)

K' = -z/(sqrt(81-z²) :: Use chain rule

0 = -z/(sqrt(81-z²)

z = 0

Step 2: Find the value of the equation when z = 0

K = 9 @ (z = 0)

Step 3: Check the domain by finding the value of K when z = ±9

K = 0 @ (z = ±9)

Looks like we have the following based off of the information we calculated:

Absolute Maximum: z = 0

Absolute Minimum: None, since they are on the edge of the domain