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How do you minimize and maximize #f(x,y)=sin^2y/x+cos^2x/y# constrained to #1<yx^2+xy^2<16#?

Answer 1

See below.

The attached plot shows the location of stationary points. The arrows represent the gradient of the objective function (black) and the boundary (red) at each stationary point. The feasible region in blue shows also level curves from the objective function. Those results were obtained using the Lagrange Multipliers technique.

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Answer 2

Ezekiel Alexander

To minimize and maximize the function ( f(x,y) = \frac{\sin^2 y}{x} + \frac{\cos^2 x}{y} ) constrained to ( 1 < yx^2 + xy^2 < 16 ), follow these steps:

Find critical points of ( f(x,y) ) by setting partial derivatives equal to zero and solving for ( x ) and ( y ).
Evaluate ( f(x,y) ) at critical points and at the boundary of the constraint region.
Determine the minimum and maximum values of ( f(x,y) ) within the constraint region.

This process involves calculus and may require solving systems of equations and optimization techniques.

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Answer from HIX Tutor

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.

How do you minimize and maximize #f(x,y)=sin^2y/x+cos^2x/y# constrained to #1&lt;yx^2+xy^2&lt;16#?

How do you minimize and maximize #f(x,y)=sin^2y/x+cos^2x/y# constrained to #1<yx^2+xy^2<16#?