Find out if $f(x,y) = x^2-xy+4y^2$ has maximum or minimum with a constraint

Question

I have a function $f(x,y)= x^2-xy+4y^2$ and the constraint: $x+y=1$ and I would like to find local and global extrema. I used the Lagrange multiplier method and found that the only critical point I am dealing with must be for $x=\frac34$ and $y= \frac14$. This is where I am a little bit stuck however. I do not really know whether this is a local or global maximum/minimum or even one of those at all. I tried finding a result with the Hessian matrix, but I did not get a result out of this. I also do not think that $\{x,y \in \mathbb R | x+y=1\}$ is compact, so I do not know whether or not my function even has a minimum or maximum in this domain.

What can I do here to finish? Any help is greatly appreciated!

Answer 1

If you write $y=1-x$ then

$$f(x)=x^2-x(1-x)+4(1-x)^2=6x^2-9x+4$$

which has a minimum at $f_{min}=\frac{5}{8}$ and it happens for $x=\frac{3}{4}$.

Answer 2

let $$f(x,y)=x^2-xy+4y^2$$ and $$y=1-x$$ thus you will have $$f(x,1-x)=x^2-x(1-x)+4(1-x)^2$$