Determine two positive numbers whose sum is 15 and the sum of whose squares is minimum.
Let the two positive numbers be a and b.
Given: a + b = 15 … 1
Also, a2 + b2 is minima
Assume, S = a2 + b2
(from equation 1)
⇒ S = a2 + (15 – a)2
⇒ S = a2 + 225 + a2 – 30a = 2a2 – 30a + 225
⇒ = 4a - 30
⇒
Since, > 0 will give minimum value of S.
4a – 30 = 0
a = 7.5
Hence, two numbers will be 7.5 and 7.5.