Exercise $3.$ Convex POLYcon.

$[10$ points$]$

A convex polygon is a polygon wherein all interior angles are less than $\backslash (180^\{\backslash$circ$\}\backslash ),$ and every line segment between two vertices remains inside or on the boundary of the polygon.

We represent a convex polygon as an array $\backslash ($ V$[1\backslash$ldots n$]\backslash )$ where:

$-$ Each element of the array represents a vertex of the polygon as a pair of coordinates $($V$[$i$].$x$,$ V$[$i$].$y$)$;

$-\backslash ($ V$[1]\backslash )$ is the vertex with the minimum $\backslash ($ x $\backslash )-$coordinate;

$-$ The vertices appear in counterclockwise order.

For simplicity, assume that the $\backslash ($ x $\backslash )-$ and $\backslash ($ y $\backslash )-$coordinates of all vertices are distinct. Design an $\backslash ($ O$(\backslash$log n$)\backslash )$ algorithm to find the point with the maximum $\backslash ($ y $\backslash )-$coordinate. Discuss the correctness of the proposed algorithm.