If $a<0$, and $f(x)=e^{ax}+e^{-ax}$ is monotonically decreasing. The interval to which x belongs.

$x < 0$

Given a < 0, and ...(i) 

$f(x)=e^{ax}+e^{-ax}$ is decreasing

$⇒f'(x)<0⇒e^{ax}-e^{-ax}<0$

$⇒a(\frac{e^{2ax}-1}{e^{ax}})<0$   ...(ii)

as from (i) $a < 0$

$⇒(e^{2ax}-1)>0⇒e^{2ax}>1$

$⇒2ax>0⇒ax>0$

$⇒x< 0(as\, a < 0)$

Thus, f(x) is monotonically decreasing if $x < 0$.