# How do you minimize and maximize #f(x,y)=2x^2-x/(2x-3y)# constrained to #1<yx^2+xy^2<16#?

See below.

With the Lagrange Multipliers technique we get the following results

Attached the feasible region and some maxima/minima points

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To minimize and maximize the function ( f(x, y) = \frac{{2x^2 - x}}{{2x - 3y}} ) constrained to ( 1 < yx^2 + xy^2 < 16 ), follow these steps:

- Find critical points by solving the system of equations formed by setting the partial derivatives of ( f(x, y) ) equal to zero and solving for ( x ) and ( y ).
- Evaluate ( f(x, y) ) at each critical point.
- Evaluate ( f(x, y) ) at the boundary points of the constraint region ( 1 < yx^2 + xy^2 < 16 ).
- Compare all the values obtained in steps 2 and 3 to determine the minimum and maximum values of ( f(x, y) ).

This approach will yield the minimum and maximum values of ( f(x, y) ) within the given constraint region.

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