This extreme value problem has a solution with both a maximum value and a minimum value. use lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y) = x2 − y2; x2 + y2 = 81 maximum value minimum value

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are -81 and 81

The function is given as:

Subtract 81 from both sides of

Using Lagrange multiplies, we have:

Substitute and

Differentiate

Equate to 0

So, we have:

Divide both sides of by -2x

Divide both sides of by 2y

The above means that:

Recall that:

When x = 0, we have:

Take square roots of both sides

When y = 0, we have:

Take square roots of both sides

To determine the critical points, we consider:

or

or

So, the critical points are:

Substitute the above values in

Considering the above values, we have:

Hence, the extreme values are -81 and 81, respectively.

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The Lagrangian is

with critical points where the partial derivatives are identically zero:

So we have four critical points to consider: (0, -9), (0, 9), (-9,0), and (9, 0). We have

So the maximum value is 81 and the minimum value is -81.