# Minimum for f(x) = x^(2n)+1/x^(2n) ?

Apr 15, 2017

$L = 6$

#### Explanation:

The function $f \left(x\right) = x + \frac{1}{x}$ has a minimum at $x = 1$ for $x > 0$

See attached plot.
In black $f \left(x\right)$
in blue $f \left({x}^{2}\right)$
in red $f \left({x}^{4}\right)$
in green $f \left({x}^{6}\right)$)

so the function

$F \left(x\right) = f \left({x}^{2}\right) + f \left({x}^{4}\right) + f \left({x}^{6}\right)$ have the minimum at the same value, for $x = 1$ and it's value is $2 + 2 + 2 = 6$