Let x be the side of the square which is to be cut off from each corner.

Then dimensions of the box are 18 - 2x, 18 - 2x and x.

Let V be the volume of the box then

**Also,**

**For maximum or minimum value V' = 0**

⇒ (18 - 2x) (- 6) = 0

⇒ x = 9

or x = 3

Neglecting x = 9

[ **∵ For x = 9,length = 18 - 2x = 18 - 2(9) = 0]**

**Therefore,** **x = 3 is to be taken.**

V'(3) = (18 - 6)(- 6) + (18 - 18) (- 2) = 72 < 0

**Thus, volume is maximum when x = 3**

∴ Length = 18 - 2x = 18 - 6 = 12 cm;

Breadth = 18 - 2x = 18 - 6 = 12cm;

Height = x = 3 cm.

**Maximum volume of the box,**

= 12 x 12 x 3 = 432cm^{3}.