a;b;c>0 and a+b+c=k min((1/(1+a^2 )) + (1/(1+b^2 )) + (1/(1+c^2 )))=?


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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 bymathdanisur last updated on 16/Jun/21

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