How do you minimize and maximize #f(x,y)=(xy)^2x+y# constrained to #1<yx^2+xy^2<16#?
See below.
There are four characteristic points characterizing local maxima/minima at
This result can be obtained using the Lagrange Multipliers technique. The attached plot shows the feasible region with the points located at the boundaries.
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To minimize and maximize ( f(x, y) = (xy)^2  x + y ) constrained to ( 1 < yx^2 + xy^2 < 16 ), follow these steps:

Use Lagrange multipliers to set up the Lagrangian: ( L(x, y, \lambda) = (xy)^2  x + y + \lambda(yx^2 + xy^2  1) )

Calculate the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ): ( \frac{\partial L}{\partial x} = 2xy^2  1 + 2\lambda yx + \lambda y^2 ) ( \frac{\partial L}{\partial y} = 2x^2y  1 + 2\lambda xy + \lambda x^2 ) ( \frac{\partial L}{\partial \lambda} = yx^2 + xy^2  1 )

Set these partial derivatives equal to zero and solve the resulting system of equations.

Evaluate the critical points obtained in step 3.

Evaluate the function ( f(x, y) ) at the critical points and at the boundaries of the constraint region ( 1 < yx^2 + xy^2 < 16 ).

Compare the values of ( f(x, y) ) obtained in step 5 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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