An m.g.f. identity forcing symmetry about 5

Suppose $X$ has c.d.f. $F$, p.d.f. $f$ and m.g.f. $M$, where

$$e^{-5t}M(t)=e^{5t}M(-t),\qquad \forall\,t\ge0 .$$

Which of the following statements is FALSE?