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

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

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).

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

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

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

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.
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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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