mathematical puzzle

Fun with Mathematics

If p and q are positive and real numbers such that p²  +  q² = 1 . What are the maximum and minumum values of (p+q)  ?

Solution:

P²  = 1 – q²  = (1 + q) (1 –q). Let  1 + q = kp  and 1-q = p/k. By solving these two equations, we have

P = 2k/(k²+ 1)  and  q = (k² – 1)/(k² + 1). If p+q is maximum  p and q must be equal or k² -2k -1 = 0 . It gives k = (1 + √2) or p = q =(1 +√2)/(2+√2) It gives p+q = √2.

If p+q is minimum the difference between them must be maximum provided p or q cannot be greater than 1. 1+ q = p² and 1- q = 1. It is possible only when p = 1 and q = 0..So the minimum value of p+q = 1