Answer
The proof is below.
Work Step by Step
We know the following equations:
$\beta = \frac{1}{V}\frac{dV}{dT}$
$\alpha = \frac{1}{S}\frac{dS}{dT}$
We take the derivative of the equation for volume to obtain:
$\frac{dV}{dT}=3S^2\frac{dS}{dT}$
$\frac{dV}{dT}=\frac{3V}{S}\frac{dS}{dT}$
$\frac{1}{V}\frac{dV}{dT}=\frac{3}{S}\frac{dS}{dT}$
Substituting the upper two equations, we find:
$\beta = 3 \alpha$
