Find the absolute maximum and minimum values of f on the set d. f(x, y) = xy2 + 2, d = {(x, y) | x ≥ 0, y ≥ 0, x2 + y2 ≤ 3}

The absolute minimum and maximum values of a function are the least and highest values, the function can take.

The absolute maximum and minimum values of f are 4 and 2, respectively.

The function is given as:

First, we determine the partial derivatives of f(x,y)

Substitute 0 for y in the first equation.

So, we have:

This means that:

Next, we calculate the extreme values.

We have:

Express as an equation

Make y the subject

Take square roots

Substitute in

So, we have:

Open brackets

Differentiate

Set to 0

Collect like terms

Divide through by 3

Take square roots of both sides

The domain is given as:

So, the possible value of x is:

Substitute 1 for x in

So, we have:

This means that, the absolute maximum is at: , while the absolute minimum is at

Substitute these values in f(x,y)

Hence, the absolute maximum and minimum values of f are 4 and 2, respectively.

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