Maximize Cistern Volume -- (cut out 4 squares (at Corners) and discard them)

On 03/05/2025  David Entwistle wrote:

Adapted from "Amusements in Mathematics" by Henry Ernest Dudeney.
>
      Given a large sheet of zinc, measuring (before cutting) one metre
square,
you are asked to cut out square pieces (all of the same size) from the
four corners of this sheet. You are then to fold up the sides of the
resulting shape, solder the edges and make a cistern. The cistern is
open
at the top.
>
     The puzzle is :   what size should the cut out pieces be, such that
the
cistern will hold the greatest possible quantity of water?

 Thanks!   (my slight editing may have introduced English Usage  errors)
___________________
        Ok... i remember High-School  Calculus now.
                   V    =   x * (1-2x)^2
                   V'   =   12 x^2  - 8 x  + 1
I let the derivative be 0 and solve it ,  and i get     x = 1/2,   1/6
            at  x=0  the slope is 1
   whereas  at  x=1/2,  the slope is Zero!!!
_______________
 at  x=1/2,  the slope is Zero!!!
         It's not obvious why,   Can someone explain this?
 at  x=0  the slope is 1  --------- I can visualize this.
                         It's like a Bean-Sprout,  coming out of the
Earth.
 at  x=1/2,  the slope is Zero!!!  ---- I can't visualize this one.