if-a-b-c-positive-real-numbers-and-a-1-a-b-1-b-c-1-c-1-prove-that-abc-1-8-

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