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## Homework Statement

Finding the stationary point(s) of the function:

[tex]f(x,y) = xy - \frac{y^{3}}{3}[/tex]

.. on the line defined by [itex]x+y = -1[/itex].

For each point, state whether it is a minimum or maximum.

## Homework Equations

.. within the problem statement and solutions.

## The Attempt at a Solution

This is what I have so far:

[tex]f(x,y) = xy - \frac{y^{3}}{3}[/tex]

[tex]g(x,y) = x+y-1 = 0[/tex]

Therefore need to extemise:

[tex]F(x,y,\lambda) = f + \lambda g = xy - \frac{y^{3}}{3} + \lambda(x+y-1)[/tex]

So calculating the partial derivatives:

[tex]\frac{\partial F}{\partial x} = y + \lambda = 0[/tex]

[tex]\frac{\partial F}{\partial y} = x - 3\left(\frac{y^{2}}{3}\right) + \lambda = x - y^{2} + \lambda = 0[/tex]

[tex]\frac{\partial F}{\partial \lambda} = x + y - 1 = 0[/tex]

Then need to look for all consistent solutions:

[tex]1. y = \lambda[/tex]

.. but now I'm stuck on what to do now, seemto have done something wrong because I can't get more consistent soluations and then nice simultaneous equations to equate