partial derivatives...minimising functions

[judjud410]

can someone please help asap

Minmise the function f(x,y)= x squared + 3y squared+ 1

subject to the constraint 1 - x - y = 0

 

[royhaas]

Substitute y=1-x and solve as one-dimensional problem.

 

[galactus]

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.