How do you maximize and minimize #f(x,y)=(xy)^2e^y+x# constrained to #0<=xy<=1#?
See below.
Using Lagrange Multipliers. Attached a plot showing the local ma xima/minima. The vectors show the objective function gradient. The stationary points are located at the feasible region frontier.
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To maximize and minimize ( f(x, y) = (xy)^2  e^y + x ) constrained to ( 0 \leq x  y \leq 1 ), we first find the critical points of ( f ) in the interior of the constraint region and on the boundary.

Interior Critical Points:
Find the partial derivatives of ( f ): [ \begin{aligned} \frac{\partial f}{\partial x} &= 2xy^2 + 1, \ \frac{\partial f}{\partial y} &= 2x^2y  e^y. \end{aligned} ] Setting these to zero gives: [ \begin{aligned} 2xy^2 + 1 &= 0, \ 2x^2y  e^y &= 0. \end{aligned} ] Solving these equations simultaneously gives the critical points. 
Boundary Critical Points:
We need to consider the boundary of the region ( 0 \leq x  y \leq 1 ) along with the constraint ( 0 \leq x, y \leq 1 ). For ( x = 0 ), ( 0 \leq y \leq 1 ), which implies ( 1 \leq y \leq 0 ).
 For ( y = 0 ), ( 0 \leq x \leq 1 ).
 For ( x = 1 ), ( 1y \leq 1 ) which implies ( 0 \leq y \leq 1 ).
 For ( y = 1 ), ( x1 \leq 1 ) which implies ( 1 \leq x \leq 2 ).
Evaluate ( f(x, y) ) at these boundary points to find the maximum and minimum values.

Final Step:
Compare the values of ( f(x, y) ) at the critical points and the boundary points to determine the maximum and minimum values of ( f ).
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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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