问题：正方形铁皮， 边长为a,截去四角，做成一个无盖的方盒，怎样截角可能做到方盒体积最大？

解答：

体积方程：v=x(a-2x)²,0<=x<=a/2

方程求导：dv/dx=(a²x+4x³-4ax²)'

     =a²-8ax+12x²

     =(a-2x)(a-6x)

在(0,a/2)内驻点为X₁=a/6,X₂=a/2;

v(a/2)=0,v(0)=0;

v(a/6)=(2/27)a³

一张A4纸大小的铁皮，截去四角，做成一个无盖的盒子，怎样截角可能做到盒子容积最大？

A4 paper size:length*width=297*210

length:297-2x

width:210-2x

volume =(297-2x)( 210-2x)x

=62370x-594x²-420x²+4x³

=4x³-1014x²+62370x

上述函数的导数：[(4x³-1014x²+62370x)]^'

=12x²-2028x+62370

12x²-2028x+62370=0

x=(-b±√(b²-4ac))/2a

√(b²-4ac)=√(4112784-2993760)=√1119024

=1057.839307267412898719450940954

X₁=128.57663780280887077997712253975(as (210-2X₁)<0,not qulified)

X₂=40.42336219719112922002287746025

任意长宽的平面w*L

length=w-2x

width=L-2x

volume =(w-2x)( L-2x)x

=wLx-2wx²-2Lx²+4x³

=4x³-2(w+L)x²+wLx

〖(4x³-2(w+L)x²+wLx)〗^'

=12x²-4(w+L)x+wL

12x²-4(w+L)x+wL =0

(W+L±√(w²+L²-wL))/6

√(b²-4ac)=√(4112784-2993760)=√1119024

=1057.839307267412898719450940954

X₁=128.57663780280887077997712253975(as (210-2X₁)<0,not qulified)

X₂=40.42336219719112922002287746025