An open-top box is to be made by cutting small congruent squares from the corners of a 9 cm × 9 cm sheet of metal and bending up the sides. what is largest possible volume of such a box

Let the length of the side of the 4 small squares be = x.

The formula for the volume of the box will be 
height * width * length

V = x ( 9 - 2x)^2

V = 81x - 36x^2 + 4x^3

finding the derivative:-

dV / d x =  12x^2 - 72x + 81

THis equals 0 for a maximum / minimum value

12x^2 - 72x + 81 = 0

3(4x^2 - 24x + 27) = 0

x = 4.5 ,  1.5 

Use second derivative to find  maxm and minm:-

d^2V / dx^2 = 24x - 72

when x = 1.5 this is negative and when x = 4.5 this is positive

so x = 1.5 gives a maximum  value for V

V = 1.5(9- 2(1-5))^2 =  54  

Largest possible volume of the box is 54 cm^3  Answer