Find the positive number x such that the sum of x and its reciprocal is as small as possible.
I'm having a bit of an issue with this one. The answer in my textbook says x=1, but I can't figure out how to get it.
I assumed that the formula to be optimized is $Q=x+\frac{1}{x}$. Taking the derivative of this gets me $Q'=-\frac{1}{x^2}$ and $Q''=\frac{2}{x^3}$
The only point where a critical value can exist on $Q'$ and $Q''$ is $x=0$... but that's not right.
Where did I go wrong here?