Let x cm be the side of square base and h cm be its height.
Then x2 + 4xh = 192
\(\therefore h = \frac{192 - x^2}{4x}....(1)\)
Let V be the volume of the box.
∴ by the second derivative test, V is maximum at x = 8.
Hence, the volume of the box is largest, when the side of square base is 8 cm and its height is 4 cm.