Solution
= Solution
Write $\phi=\Phi'$ for the standard <normal density>. Differentiating the <Black-Scholes formula> and using $S_0\phi(d_1)=Ke^{-rT}\phi(d_2)$ gives
$$
\frac{\partial C}{\partial\sigma}
=S_0\phi(d_1)\left(\frac{\partial d_1}{\partial\sigma}-\frac{\partial d_2}{\partial\sigma}\right)
=\boxed{S_0\sqrt T\,\phi(d_1)>0}.
$$
The last equality follows by differentiating $d_1-d_2=\sigma\sqrt T$. This derivative is the option's <vega>.