# How do you minimize and maximize #f(x,y)=(x-y)(x+y)+sqrt(xy)# constrained to #0<xy-y^2<5#?

The points

We will searching for stationary points, qualifying then as local maxima/minima.

First we will transform the maxima/minima with inequality restrictions into an equivalent maxima/minima problem but now with equality restrictions.

To do that we will introduce the so called slack variables

Maximize/minimize

constrained to

The lagrangian is given by

The condition for stationary points is

so we get the conditions

Solving for

The restriction

Computing

and

we conclude that the found solutions are local minima points.

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To minimize and maximize the function ( f(x, y) = (x - y)(x + y) + \sqrt{xy} ) constrained to ( 0 < xy - y^2 < 5 ), follow these steps:

- Find critical points of ( f(x, y) ) by setting partial derivatives with respect to ( x ) and ( y ) equal to zero.
- Evaluate the function at critical points and boundary points of the constraint region.
- Compare the values obtained in step 2 to determine the minimum and maximum values of ( f(x, y) ) within the constraint region.

The critical points can be found by solving the system of equations:

[ \frac{\partial f}{\partial x} = 2x - 2y + \frac{1}{2\sqrt{xy}}(y + 2x) = 0 ] [ \frac{\partial f}{\partial y} = -2x + 2y + \frac{1}{2\sqrt{xy}}(x - 2y) = 0 ]

Solve these equations simultaneously to find the critical points. Then evaluate ( f(x, y) ) at these points and at the boundary points of the constraint region ( 0 < xy - y^2 < 5 ). Compare the values obtained to determine the minimum and maximum values of ( f(x, y) ) within the constraint region.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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