# How do you maximize and minimize #f(x,y)=x^2+9y^2-xy# constrained to #3<xy<5#?

If your constraints are as written, then

The partial derivatives of

Therefore, the maximum and minimum values of

The Method of Lagrange Multipliers can now be used. The gradient vector of

We then have

If you now look at the contour map of the level curves (shown further below...dark means smaller outputs and light means higher outputs), you'll see that a global minimum for

A local minimum value for

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To maximize and minimize ( f(x,y) = x^2 + 9y^2 - xy ) constrained to ( 3 < xy < 5 ), you first need to find the critical points of the function within the constraint.

- Find the critical points of ( f(x,y) ) by taking partial derivatives with respect to ( x ) and ( y ) and setting them equal to zero.
- Solve for ( x ) and ( y ) from the equations obtained in step 1.
- Check the critical points to ensure they lie within the constraint ( 3 < xy < 5 ).
- Evaluate the function ( f(x,y) ) at each critical point to determine the maximum and minimum values.

However, since the constraint region is non-rectangular, finding the critical points and analyzing the function's behavior becomes more complex and might require numerical methods or graphical analysis.

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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.

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.

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.

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