What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

Use the diagram to investigate the classical Pythagorean means.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.