How do you minimize and maximize #f(x,y)=ye^xxe^y# constrained to #xy=4#?
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To minimize and maximize ( f(x,y) = ye^x  xe^y ) constrained to ( xy = 4 ), we can use the method of Lagrange multipliers.

Form the Lagrangian function: ( L(x, y, \lambda) = ye^x  xe^y + \lambda(xy  4) )

Compute the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ) and set them equal to zero: [ \frac{\partial L}{\partial x} = ye^x  e^y + \lambda y = 0 ] [ \frac{\partial L}{\partial y} = e^x  xe^y + \lambda x = 0 ] [ \frac{\partial L}{\partial \lambda} = xy  4 = 0 ]

Solve the system of equations simultaneously to find critical points.

Substitute the critical points into ( f(x,y) ) and evaluate them to determine the minimum and maximum values.
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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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