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
⇒ \(\frac{ds}{da}\) = 4a - 30
⇒ \(\frac{d^2s}{da^2}\) = 4
Since,
\(\frac{d^2s}{da^2}\) > 0
⇒ \(\frac{ds}{da}\) = 0 will give minimum value of S.
4a – 30 = 0
⇒ a = 7.5
Hence, two numbers will be 7.5 and 7.5.
