a-b-c-gt-0-and-a-b-c-k-min-1-1-a-2-1-1-b-2-1-1-c-2-

Question Number 143591 by mathdanisur last updated on 16/Jun/21

a; b; c > 0 a n d a + b + c = k

m i n (\frac{1}{1 + a^{2}} + \frac{1}{1 + b^{2}} + \frac{1}{1 + c^{2}}) = ?

Answered by Olaf_Thorendsen last updated on 16/Jun/21

Let f (a, b, c) = \frac{1}{1 + a^{2}} + \frac{1}{1 + b^{2}} + \frac{1}{1 + c^{2}}

m i n (f (a, b, c)) = ?

By symmetry a = b = c = \frac{k}{3}

m i n (f (a, b, c)) = \frac{3}{1 + {(\frac{k}{3})}^{2}}

m i n (f (a, b, c)) = \frac{27}{9 + k^{2}}

Commented by mathdanisur last updated on 16/Jun/21

t h a n k s S i r, t h e r e a r e a l s o t w o

o t h e r r o o t s . .