# How do you minimize and maximize #f(x,y)=(e^(yx)-e^(-yx))/(2yx)# constrained to #1<x^2/y+y^2/x<3#?

Twho local maxima at

Completing with the so called slack variables

Minimize/maximize

subjected to

The lagrangian reads

The lagrangian stationary points includes the local maxima/minima points.

The stationary points are the solutions of

or

Solving for

There are four solutions but for two points. The first

and

Calculating

Attached the figure with the feasible region showing the

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To minimize and maximize ( f(x,y) = \frac{e^{yx} - e^{-yx}}{2yx} ) subject to ( 1 < \frac{x^2}{y} + \frac{y^2}{x} < 3 ), we can use the method of Lagrange multipliers.

Define the Lagrangian function ( L ) as:

[ L(x, y, \lambda) = \frac{e^{yx} - e^{-yx}}{2yx} + \lambda \left( \frac{x^2}{y} + \frac{y^2}{x} - 3 \right) ]

Taking partial derivatives with respect to ( x ), ( y ), and ( \lambda ) and setting them to zero will give us the critical points.

- Partial derivative with respect to ( x ):

[ \frac{\partial L}{\partial x} = \frac{y(e^{yx} + e^{-yx})}{2yx^2} + \lambda \left( \frac{2x}{y} - \frac{y^2}{x^2} \right) = 0 ]

- Partial derivative with respect to ( y ):

[ \frac{\partial L}{\partial y} = \frac{x(e^{yx} - e^{-yx})}{2y^2x} + \lambda \left( \frac{x^2}{y^2} + \frac{2y}{x} \right) = 0 ]

- Partial derivative with respect to ( \lambda ):

[ \frac{\partial L}{\partial \lambda} = \frac{x^2}{y} + \frac{y^2}{x} - 3 = 0 ]

Solve this system of equations to find the critical points of ( f(x,y) ). After finding these critical points, evaluate ( f(x,y) ) at each point and compare the values to determine the minimum and maximum values of ( f(x,y) ) within the given constraint.

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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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