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How do you minimize and maximize #f(x,y)=x/e^(x-y)+y# constrained to #1<x^2/y+y^2/x<9#?

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

Emilia Aguilar

To minimize and maximize ( f(x,y) = \frac{x}{e^{(x-y)}} + y ) constrained to ( 1 < \frac{x^2}{y} + \frac{y^2}{x} < 9 ), you can follow these steps:

Find the critical points of ( f(x,y) ) by taking partial derivatives with respect to ( x ) and ( y ) and setting them equal to zero.
Test the critical points along with the boundary of the constraint region to determine which points give the minimum and maximum values of ( f(x,y) ).
Use Lagrange multipliers if necessary to handle the constraint ( 1 < \frac{x^2}{y} + \frac{y^2}{x} < 9 ).
Calculate the values of ( f(x,y) ) at the critical points and along the boundary to find the minimum and maximum values.

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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)=x/e^(x-y)+y# constrained to #1&lt;x^2/y+y^2/x&lt;9#?

How do you minimize and maximize #f(x,y)=x/e^(x-y)+y# constrained to #1<x^2/y+y^2/x<9#?