Statement: if $c(t)$ is the price of the digital cash-or-nothing call option, then direct calculation (under Black-Scholes assumptions) shows that $$\frac{\partial c(t))}{\partial \sigma^2 }>0 \quad\text{if and only if}\quad S(t)<Xe^{-(r+\frac{1}{2} \sigma^2 )(T-t)}.$$
I fail to prove this statement (I do not even know how to start).
Can anyone give me some hints to proceed?