Minimisation problem on cuboid

Question

A problem asks to find the shortest path on a cuboid with volume $1$ connecting two opposite corners. How does one solve this? I think I have to minimise $(a+b)^2+c^2$ with $abc = 1$ but I have no idea how to continue from this.

Answer 1

Using the AM-GM inequality, we have $$(a+b)^2 \geqslant 4ab = \frac{4}{c},$$ so $$(a+b)^2+c^2 \geqslant \frac{4}{c} + c^2 = \frac{2}{c} + \frac{2}{c} + c^2\geqslant 3\sqrt[3]{\left(\frac{2}{c}\right)^2 \cdot c^2} = 3\sqrt[3]{4}.$$ Equality hold for $a = b = \frac{1}{\sqrt[6]{2}}, \, c = \sqrt[3]{2}.$

Therefore, the minimum value is $3\sqrt[3]{4}.$