A box open from top is made from a rectangular sheet of dimension a...

Given that: A box open from top is made from a rectangular sheet dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps and the volume of the box is maximum.

dimensions of the box:

$L = a - 2x, B = b - 2x \text{ and } H = x$

$V = L \times B \times H$

$V ( x ) = ( a - 2x ) ( b - 2x ) x$

$V ( x ) = 4{x}^{3} - 2 ( a + b ) {x}^{2} + abx$

For volume to be maximum:
${V}^{'} ( x ) = 12{x}^{2} - 4 ( a + b ) x + ab = 0$

$x = \dfrac{4 ( a + b ) - \sqrt{16{a}^{2} + 16{b}^{2} - 16ab}}{24}$$\left[+ cannotbetakenasx < a, b\right]$,
for maximum volume, $x = \dfrac{a + b - \sqrt{{a}^{2} + {b}^{2} - ab}}{6}$