What is the minimum value of X + 1/X=?
Let’s assume further that x is positive real.
We start with a simple fact that ANY squares are non-negative. The idea is we “choose” a square intelligently to suit our purpose. Also not that x does NOT equal zero.
I choose the square ( x^(1/2) – 1 / [ x^(1/2)] )^2.
It must be non-negative for whatever positive real values of x.
( x^(1/2) – 1 / [ x^(1/2)] )^2 >= 0
Expanding and regrouping, we get,
x – 2 + 1/x >= 0
x + 1/x >= 2
Thus, minimum is 2.
With equality when x^(1/2) = 1 / [ x^(1/2)], or
x = 1.
One can use two variable AM – GM,
For any two positive reals m and n, we have
( m + n )/ 2 >= (mn)^(1/2)
With equality if and only if m = n.
Just set m = x and n = 1/x, to get,
( x + 1/x )/ 2 >= 1
or x + 1/x >= 2
Again, the minimum values is 2 with equality if and only if x = 1 .