Calculus Question

Question

A box is constructed from a square piece cardboard with sides of 4 feet by cutting equal squares from the corners and folding up the sides as shown. The volume of the box is given by: V=4x^3-16x^2+16x. It is determined that the volume will be at a maximum if the cut out squares have sides of 2/3ft. Is this correct? Explain why or why not.

Answer 1

V = x(4 - 2x)² = 4x³ - 16x² + 16x; V' = 12x² - 32x + 16 = 0; 3x² - 8x + 4 = 0; x = (4 ± 2)/3; x = 2 not make box; or x = 2/3

V'' = 24x - 32; v''(2/3) = 16 - 32 = - 16 < 0, so, at x = 2/3 we have max volume box