My child's teacher raised a quesion in class for students who are interested to prove. The teacher says that the volume of a cube is the greatest among rectangular-faced shapes of the same perimeter and asks his students to prove this proposition.
I considered the relationship between the length of the sides of a cube and the lengths of the sides of rectangular-faced shapes in different situation. But when the calculations came down to polynomials, I couldn't proceed due to the uncertainty of the variables in the polynomials.
Can anyone please find a good way to prove the above proposition? Or is there already a proof? Thank you for your help!
Best Answer
Is elementary solutions permitted?
$$ \frac{a+b+c }{3}\geq \sqrt[3]{abc} $$
Equality i.e. maximum volume for a given sum of side lengths is when all sides are equal