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fix any arbitary point, find another point that results in the minimum/maximum slope

Graham scan algorithm.

Graham scan could be used to find convex hull.

Create equation of the line by selected two points.
Check every other point by this equation:
if result >0 then point at one side of the line
if result < 0 then point at another side of the line
if result == 0 the point belongs to line.

Absolutely right Computational Geometry, Thats Convex Hull. Crazy Question man

public static Point[] func(Point[] points) {
        if (points == null || points.length <= 1) return null;
        Point[] answer = new Point[2];
        int length = points.length;
        Point firstPoint = points[0];
        int j = 0;
        for (int i = 1; i < length; i++) {
            Point tempPoint = points[i];
            if (tempPoint.x < firstPoint.x) {
                firstPoint = tempPoint;
                j = i;
            }
        }
        answer[0] = firstPoint;
        answer[1] = points[(j+1)%length];
        for (int i = 0; i < length; i++) {
            boolean flag = checkDirection(answer[0], points[i], answer[1]);
            if (flag) {
                answer[1] = points[i];
            }
        }
        return answer;
}

private static boolean checkDirection(Point p1, Point p2, Point p3) {
        int value = ((p3.y - p2.y)*(p2.x-p1.x)) - ((p2.y - p1.y)*(p3.x-p2.x));
        if (value > 0) return true;
        return false;
}

static class Point {
        int x;
        int y;
        Point (int xx, int yy) {
            x = xx;
            y = yy;
        }
}

Only the first step of a graham scan is needed. Scan once for the lowest x co-ordinate and find another point with the largest angle to that point.