Find the absolute maxima and minima of the function on the given domain.
-$$f ( x , y ) = x ^ { 2 } + y ^ { 2 } \text { on the diamond-shaped region } | x | + | y | \leq 4$$
A) Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 0 at (0, 0)
B) Absolute maximum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 ; absolute minimum: 0 at (0, 0)
C) Absolute maximum: 16 at (4, 0) and (0, 4); absolute minimum: 0 at (-4, 0) and (0, -4)
D) Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 ,

Question

Quizplus · Accepted Answer

The function $f(x, y) = x^2 + y^2$ reaches its maximum on the boundary of the given domain $|x| + |y| \leq 4$ when either $x$ or $y$ is maximized, which happens at the points (4, 0), (-4, 0), (0, 4), and (0, -4), giving a maximum value of $16$. The minimum value of $f$ is $0$, which occurs at the origin (0, 0), where both $x$ and $y$ are $0$.

Find the Absolute Maxima and Minima of the Function on the Given