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

##### 2 Answers

Constrain even more to

#### Explanation:

Suppose we constrain

#f(x, y) = x^2+y^3 = (1-3y)^2+y^3 = y^3+9y^2-6y+1#

The end behaviour is dominated by the

#lim_(y->+oo) f(x, y) = +oo#

#lim_(y->-oo) f(x, y) = -oo#

See below.

#### Explanation:

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.

Ex.

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 points showing the gradient direction.

The qualification is left to the reader.

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