- The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by
$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$
- Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
- Hence find the maximum volume of the box.
- Use calculus to justify that the volume that you found in part (b) is a maximum.