A recent paper (the context is not important) wants to find the minimum of
the function:
y(x) = sqrt(4+4x) - sqrt(x).
Using calculus, one finds x=1/3 and thus y=sqrt(3).
The question is: Is there a neat way to find this solution without using
calculus? I don't want an approximation, which you can do graphically or
numerically. Nor do I think a Taylor expansion around x=1/3 is really fair.
I'm seeking a geometric solution. Rearrange the equation into a parabola or
some such and hence show the minimum must be at x=1/3. I've managed to
rearrange the equation into quite a variety of forms (as you can easily
amuse yourself doing) but so far they haven't given me the insight I'm
looking for. I've also tried a bunch of different variable substitutions.
Anyone see some way? Again, what is wanted is a convincing and clear
approach. If it's too fancy, forget it. I somehow feel sure there's got to
be a simple way to do it. -Carl