# How do you minimize and maximize #f(x,y)=(x-y)/((x-2)^2(y-4))# constrained to #xy=3#?

Local maximum located at

We will be looking for stationary points with posterior qualification.

This technique consists in finding points such that the normal vector to the objective function

and the restriction function

are aligned. Formally speaking, there exists

proceeding this way we obtain the equation set

Without much effort can be obtained the unique real solution

Qualifying this point must be done considering the restriction so the easiest way to do that is substituting the restriction

objective function

then we can verify that

and

qualifying this point as a local maximum.

Attached are two figures. One representing the surface intersection of

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To minimize and maximize ( f(x,y) = \frac{x - y}{(x - 2)^2(y - 4)} ) constrained to ( xy = 3 ), you can use the method of Lagrange multipliers.

First, write the Lagrangian:

[ L(x, y, \lambda) = \frac{x - y}{(x - 2)^2(y - 4)} + \lambda(xy - 3) ]

Next, find the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ), and set them equal to zero to find critical points.

[ \frac{\partial L}{\partial x} = \frac{y(x - 4) + \lambda y}{(x - 2)^2(y - 4)^2} = 0 ]

[ \frac{\partial L}{\partial y} = \frac{x(x - 4) - \lambda x}{(x - 2)^2(y - 4)^2} = 0 ]

[ \frac{\partial L}{\partial \lambda} = xy - 3 = 0 ]

Solve these equations simultaneously to find the critical points. Then, evaluate ( f(x, y) ) at each critical point and compare the values to find the minimum and maximum.

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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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