I need to find the maximum value of the $$f(x,y) = x^2 + 2y^2$$ subject to the constraint $$y-x^2+1=0$$
Now I know this problem can be solved via lagrange multipliers and I have got the maximum value as $2$.
But If i try to substitute the value of $$x^2 =y+1$$ then I get $$f = 2y^2 + y + 1$$ and clearly for this function No maximum value exist.
My question is: Why does substitution fails to give any maxima\minima for given function, also for any similar problems which involve some constraint, Do I always need to use Lagrange multipliers only ?
Thank you.