How do you minimize and maximize #f(x,y)=(x-y)/(x^2+y^2)# constrained to #0

Question

How do you minimize and maximize #f(x,y)=(x-y)/(x^2+y^2)# constrained to #0<xy-y^2<5#?

Isabella Ackerman · Accepted Answer

See below.

Local maxima/minima can be located using the Lagrange Multipliers technique. The attached figure shows the two stationary points with local maxima/minima, at the feasible region border. The arrows show the objective function gradient direction at those points.

#((f(x,y),x,y),(-0.137852,-5.44061,-1.17109),(0.137852,5.44061,1.17109))#

Emma Aldridge · Answer

undefined To minimize and maximize the function $f(x,y) = \frac{{x - y}}{{x^2 + y^2}}$ subject to the constraint $0 < xy - y^2 < 5$, we can use the method of Lagrange multipliers.

1. Form the Lagrangian function: $L(x, y, \lambda) = \frac{{x - y}}{{x^2 + y^2}} + \lambda(xy - y^2 - 5)$.

2. Compute the partial derivatives of $L$ with respect to $x$, $y$, and $\lambda$, and set them equal to zero:

$\frac{{\partial L}}{{\partial x}} = \frac{{y^2 - 2x(x - y)}}{{(x^2 + y^2)^2}} + \lambda(y) = 0$,

$\frac{{\partial L}}{{\partial y}} = \frac{{-x^2 + 2y(x - y)}}{{(x^2 + y^2)^2}} + \lambda(x - 2y) = 0$,

$\frac{{\partial L}}{{\partial \lambda}} = xy - y^2 - 5 = 0$.

3. Solve this system of equations to find critical points $(x, y, \lambda)$.

4. Test each critical point using the second derivative test to determine whether it corresponds to a minimum, maximum, or saddle point.

5. Evaluate $f(x,y)$ at each critical point and check if they lie within the constraint $0 < xy - y^2 < 5$.

6. Compare the values of $f(x,y)$ at the critical points to find the minimum and maximum values.

Following these steps will determine the minimum and maximum values of $f(x,y)$ subject to the given constraint.

How do you minimize and maximize #f(x,y)=(x-y)/(x^2+y^2)# constrained to #0&lt;xy-y^2&lt;5#?

How do you minimize and maximize #f(x,y)=(x-y)/(x^2+y^2)# constrained to #0<xy-y^2<5#?