Problem 3(25 pts) In this problem we study several optimization problems. You need to establish mathematical model and find extrema of your function by analyzing derivatives.

(i) Find the maximal area of rectangle inscribed in a circle of radius R. What's the shape of rectangle when it reaches maximal area?

(ii) Find the minimal vertical and horizontal distance between graph of y = 2x + 3 and y = ln x.

(iii) Prove Reflection Law for Optics: Angle of incidence equals angle of reflection. (Hint: Model this problem similar to Refraction Principle we derived in class).