## Maxima and Minima using calculus

**Problem:**

The vertices of an -gon lie on the sides of a regular n-gon and divide its perimeter into parts of equal length. How should one construct the gon so that its area is :

(a) maximum

(b) minimum

**Hint only:**

[One of the golden rule of solving problems in math/physics is to draw diagrams, as had benn emphasized by the maverick American physics Nobel Laureate, Richard Feynman. He expounded this technique even in software development. So, in the present problem, first draw several diagrams.]

There exists a side of the -gon that lies entirely on a side of the n-gon. Let and . Show that . Then, for , we have and the area S of the -gon is given by

where . Thus, is a quadratic function of x. Show that is a minimal when or and is maximal when .

*Let me know if you have any trouble when you **attempt it,*

-Nalin Pithwa

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