If the length of a side of the bottom is x and the height is h, then the volume of the box is given by,
`V = x^2h`
4 = x^2h
Therefore, `h = 4/x^2`
The area of the material needed is A,
`A = x^2+4xh`
`A = x^2+4x(4/x^2)`
`A = x^2+16/x`
To find the extreme points (maxima, minima or inflection points) we have to check for the points where the first derivative is zero.
`(dA)/(dx) = 2x-16/x^2`
At extreme points, (dA)/(dx) = 0
`2x-16/x^2 = 0`
`x^3 = 8`
`x = 2`
Therefore at x=2, we have an extreme point to determine whether this is a minimum point we have to evaluate the sign of the second derivative.
`(d^2A)/(dx^2) = 2+32/x^3`
This is positive for every x >0, then we can say that at x=2 , A has a minimum value.
The minimum area, `A_min= 2^2+16/2 = 12`
The minimum area is 12.
The height corresponding to minimum area, h is,
`h = 4/2^2 = 1`
The dimensions for minimum amount of materials are width of 2 and height of 1, x=2 and h =1.