From the figure:

$D^2 = d^2 + h^2$

$0 = 2d + 2h \dfrac{dh}{dd}$

$\dfrac{dh}{dd} = -\dfrac{d}{h}$

Volume of cylinder:

$V = \frac{1}{4}\pi d^2 h$

$\dfrac{dV}{dh} = \dfrac{\pi}{4} \left[ d^2 \dfrac{dh}{dd} + 2dh \right] = 0$

$d \dfrac{dh}{dd} + 2h = 0$

$d\left( -\dfrac{d}{h} \right) + 2h = 0$

$2h = \dfrac{d^2}{h}$

$d^2 = 2h^2$

$d = \sqrt{2} \, h$

$\text{diameter } \, = \sqrt{2} \, \times \, \text{ height }$ *answer*