how to solve the "best box" problem

can anyone please explain, how to solve this problem.

See, it can be easily proved that a square has the largest area among all rectangles with given perimeter and area.
Still, proof: A=xy and P=2(x+y).
Replacing y with x in A, we get A=x(P/2-x). Differentiating w.r.t. x we get P/2=2x. Putting this is equation of P we get x=y.
Now, getting back to the question, given l=4a+4b+4c or l/4 = a+b+c. And S=2(ab + bc + ca) or S/2=a(l/4-a)+bc. Now multiply a both sides to get V (volume) in terms of a. So, we get S*a/2= a * a (l/4 - a) + V.
Finally we get V in terms of a. Now differentiate it and get the two values of a from the quadratic equation. One value will occur once and the other value will occur twice. Since two sides are equal for maximum surface area. Just check which value has to occur twice from the perimeter equation, and find the volume!

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