Find the absolute maximum and minimum values of the function f(x, y) = x2 + xy + y2 on the disc x2 + y2 ≤ 1. (you do not have to use calculus.)

The required  absolute maximum and minimum values of the function is 0 and .

Given that,

Function on disc .

We have to determine,

The absolute maximum and minimum values of the function.

According to the question,

Consider the polar points,

The equation of the circle,

Then, (x, y) can be written as,

Therefore,   which means they share the same extreme values.

Over the disk, points lies

Where, r = 0 then

And Where r = 1 then,

When

Then,

Along the boundary we get

Therefore, f(x, y) has a minimum value of 0 and a maximum value of over the unit disk .

Hence, The required  absolute maximum and minimum values of the function is 0 and .

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