# How do you minimize and maximize #f(x,y)=x^2-y/x# constrained to #0<x+3y<2#?

See below.

Given

We are looking for local minima/maxima.

This problem can be handled using lagrange multipliers. This can be done following the steps:

1) Describe

This can be done introducing the so called slack variables

then

2) Form the Lagrangian

3) Determine the stationary points of

This is done computing the solution to

where

so the equations which define the stationary points are

Solving for

4) Qualifying the stationary points

This can be done computing

and depending on the sign, if positive it is a local minimum and if negative a local maximum.

In our case,

The evaluations must be done according to the values found for

Attached a plot showing the region with the objective function level curves, and the stationary point.

The qualification is left to the reader. I think it is a local minimum.

Note. Of course if the evaluation gives zero we will continue the qualification process but this is another chapter.

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The intent may have been to apply derivatives, but the explanation below would seem to provide a simpler solution:

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To minimize and maximize ( f(x, y) = x^2 - \frac{y}{x} ) constrained to ( 0 < x + 3y < 2 ), you need to find the critical points of the function within the constraint region and then evaluate the function at these points to determine the minimum and maximum values. This involves finding the partial derivatives of ( f(x, y) ) with respect to ( x ) and ( y ), setting them equal to zero to find critical points, and then checking the boundary of the constraint region.

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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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