J judjud410 New member Joined Apr 30, 2006 Messages 5 Jun 5, 2006 #1 can someone please help asap Minmise the function f(x,y)= x squared + 3y squared+ 1 subject to the constraint 1 - x - y = 0
can someone please help asap Minmise the function f(x,y)= x squared + 3y squared+ 1 subject to the constraint 1 - x - y = 0
R royhaas Full Member Joined Dec 14, 2005 Messages 832 Jun 5, 2006 #2 Substitute y=1-x and solve as one-dimensional problem.
G galactus Super Moderator Staff member Joined Sep 28, 2005 Messages 7,203 Jun 5, 2006 #3 A more 'around-the-horn' approach could be Lagrange multipliers. \(\displaystyle \L\\(x^{2}+3y^{2}+1)dx=2x\) \(\displaystyle \L\\(x^{2}+3y^{2}+1)dy=6y\) \(\displaystyle \L\\2xi+6yj={\nabla}f\) \(\displaystyle \L\\-i-j={\nabla}g\) \(\displaystyle \L\\2xi+6yj={\lambda}(-i-j)\) \(\displaystyle \L\\2x={-\lambda}i\rightarrow{x=\frac{-\lambda}{2}}\) \(\displaystyle \L\\6yj={-\lambda}j\rightarrow{y=\frac{-\lambda}{6}}\) \(\displaystyle \L\\{-}2x={\lambda}\) and \(\displaystyle \L\\{-}6y={\lambda}\) \(\displaystyle \L\\2x=6y\) Now, do that 1-x thing royhass mentioned and see if you get the same answer.
A more 'around-the-horn' approach could be Lagrange multipliers. \(\displaystyle \L\\(x^{2}+3y^{2}+1)dx=2x\) \(\displaystyle \L\\(x^{2}+3y^{2}+1)dy=6y\) \(\displaystyle \L\\2xi+6yj={\nabla}f\) \(\displaystyle \L\\-i-j={\nabla}g\) \(\displaystyle \L\\2xi+6yj={\lambda}(-i-j)\) \(\displaystyle \L\\2x={-\lambda}i\rightarrow{x=\frac{-\lambda}{2}}\) \(\displaystyle \L\\6yj={-\lambda}j\rightarrow{y=\frac{-\lambda}{6}}\) \(\displaystyle \L\\{-}2x={\lambda}\) and \(\displaystyle \L\\{-}6y={\lambda}\) \(\displaystyle \L\\2x=6y\) Now, do that 1-x thing royhass mentioned and see if you get the same answer.
