How do you maximize and minimize #f(x,y)=xxy^2# constrained to #0<=x^2+y<=1#?
There are several local maxima/minima
With the so called slack variables
Now the lagrangian formulation reads:
minimize/maximize
subjected to
forming the lagrangian
The local minima/maxima points are included into the lagrangian stationary points found by solving
or
Solving for
Those five points must be qualified. The first second and thirt activate constraint
giving
Attached a figure with the
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To maximize and minimize ( f(x,y) = x  xy^2 ) constrained to ( 0 \leq x^2 + y \leq 1 ), you can use the method of Lagrange multipliers. Here are the steps:

Set up the Lagrangian function: [ L(x, y, \lambda) = x  xy^2 + \lambda(x^2 + y  1) ]

Compute the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ), and set them equal to zero: [ \frac{\partial L}{\partial x} = 1  y^2 + 2\lambda x = 0 ] [ \frac{\partial L}{\partial y} = 2xy + \lambda = 0 ] [ \frac{\partial L}{\partial \lambda} = x^2 + y  1 = 0 ]

Solve the system of equations obtained in step 2 to find the critical points.

Check the points obtained in step 3 to determine which points satisfy the given constraint ( 0 \leq x^2 + y \leq 1 ).

Evaluate ( f(x, y) ) at each critical point satisfying the constraint to find the maximum and minimum values.

Compare the values obtained to find the maximum and minimum 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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