- #1
Hart
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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