A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each of side…

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:

$L=a-2x, B=b-2x \& H=x$

$V=L\times B\times H$

$V\left(x\right)=(a-2 x)(b-2 x) x$

$V\left(x\right)=4x^3-2(a+b)x^2+abx$

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

$x=\frac{4(a+b)-\sqrt{16a^2+16b^2-16ab}}{24}$$\left[+ \mathrm{cannot} \mathrm{be} \mathrm{taken} \mathrm{as} x<a,b\right]$,
for maximum volume, $x=\frac{a+b-\sqrt{a^2+b^2-ab}}{6}$