How do you minimize and maximize #f(x,y)=(xy)/x^2# constrained to #xy=4#?
simplest way here is to convert this to a problem in single variable calculus
so the fixed points are
the advantage of single var over Lagrange Multipe is that the nature of the fixed points are found mechanically via the second derivative
this is consistent with evaluation of the function at these points
further work on the Hessian matrix might reveal the nature of the fixed points but that is already apparent from the single variable approach
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To minimize and maximize the function ( f(x,y) = \frac{xy}{x^2} ) constrained to ( xy = 4 ), we can use the method of Lagrange multipliers.

Set up the Lagrangian: [ L(x, y, \lambda) = \frac{x  y}{x^2} + \lambda(xy  4) ]

Find the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ), and set them equal to zero: [ \frac{\partial L}{\partial x} = \frac{2x^2  y + 2xy}{x^3} + \lambda y = 0 ] [ \frac{\partial L}{\partial y} = \frac{1}{x^2}  \frac{1}{x^2}  \lambda x = 0 ] [ \frac{\partial L}{\partial \lambda} = xy  4 = 0 ]

Solve the system of equations to find critical points.

Evaluate the function ( f(x, y) ) at the critical points and the boundary points to find the minimum and maximum values.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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