Answer
$x = 0$
Work Step by Step
We can use superposition to find the equation for the standing wave:
$y'(x,t) = (6.0~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x+(8.00~s^{-1})~t] + (6.0~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x-(8.00~s^{-1})~t]$
$y'(x,t) = (12~cm)~cos~\frac{\pi}{2}[(2.00 ~m^{-1})x]~\cdot cos~\frac{\pi}{2}[(8.00~s^{-1})~t]$
We can find the values of $x$ such that $\vert cos~\frac{\pi}{2}[(2.00 ~m^{-1})x] \vert = 1$:
$\vert cos~\frac{\pi}{2}[(2.00 ~m^{-1})x] \vert = 1$
$\vert cos~[(\pi m^{-1})x] \vert = 1$
$(\pi m^{-1}) x = 0, \pi, 2\pi, 3\pi, ...$
$x = 0, 1.00~m, 2.00 ~m, 3.00~m ...$
The location of the antinode having the smallest value of $x$ is $~~x = 0$
