A candy box is made from a piece of cardboard that measures

A candy box is made from a piece of cardboard that measures 25 by 14 inches. Squares of equal size will be cut out of each corner. The sides will then be folded a rectangular box. What size square should be cut from each corner to obtain maximum volume? A square with a side of length inches should be cut away from each corner to obtain the maximum volume. (Round to the nearest hundredth as needed.)

Solution

Let the size of the square be x inches. Then the volume of the rectangle is
V=(25-2x)(14-2x)x
to be maximum dV/dx=0 and d²V/dx²<0
dV/dx=-2x(39-4x)+(25-2x)(14-2x)=0
x=10.117,2.883
d²V/dx²=-2(39-4x)+8x-2(39-4x)

=24x-156

this is <0 for x=2.883

so x=2.883