Let-a-b-c-0-c-3-a-b-3-0-and-a-2-b-2-c-2-3-Prove-that-1-2-a-3-b-c-3-b-3-c-a-3-c-3-a-b-3-2-Determine-when-equality-holds-

Question Number 145198 by loveineq last updated on 03/Jul/21

Let a ⩾ b ⩾ c ⩾ 0, c^{3} + {(a + b)}^{3} \neq 0 and a^{2} + b^{2} + c^{2} = 3 . Prove that

\frac{1}{2} ⩽ \frac{a^{3} + {(b + c)}^{3} + b^{3} + {(c + a)}^{3}}{c^{3} + {(a + b)}^{3}} ⩽ 2

Determine when equality holds .

Answered by mitica last updated on 03/Jul/21

u^{3} + v^{3} = (u + v) (u^{2} - u v + v^{2});

\Leftrightarrow \frac{1}{2} ⩽ \frac{6 + a c + b c - 2 a b}{3 + 2 a b - a c - b c} ⩽ 2 ∙

y = 2 a b - a c - b c \Rightarrow y = 2 a \cdot b - (a + b) \cdot c

2 a ⩾ a + b; b ⩾ c \Rightarrow y ⩾ 0

y ⩽ a^{2} + b^{2} - c (a + b) = 3 - c^{2} - c (a + b) = 3 - c (a + b + c) ⩽ 3

∙ \Leftrightarrow \frac{1}{2} ⩽ \frac{6 - y}{3 + y} ⩽ 2 \Leftrightarrow 0 ⩽ y ⩽ 3

Commented by loveineq last updated on 03/Jul/21

t h a n k s