if a;b;c positive real numbers and (a/(1+a)) + (b/(1+b)) + (c/(1+c)) = 1 prove that: abc ≤ (1/8)


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Question Number 151052 by mathdanisur last updated on 17/Aug/21

$if a; b; c positive real numbers and$ $\frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c} = 1 prove that :$ $abc ⩽ \frac{1}{8}$
	Answered by dumitrel last updated on 18/Aug/21

	$3 r + 2 q + p = 1 + p + q + r \Rightarrow 2 r + q = 1$ $q^{3} ⩾ 27 r^{2} \Rightarrow {(1 - 2 r)}^{3} ⩾ 27 r^{2} \Rightarrow$ $8 r^{3} + 15 r^{2} + 6 r - 1 ⩽ 0 \Rightarrow (8 r - 1) (r^{2} + 2 r + 1) ⩽ 0 \Rightarrow r ⩽ \frac{1}{8}$
	Commented by mathdanisur last updated on 18/Aug/21

	$nice Ser thank you$
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