Let-a-b-c-gt-0-and-a-b-b-c-4-Prove-that-2a-b-a-b-b-2c-b-c-8-1-2-a-2b-c-c-a-Determine-when-equality-holds-

Question Number 144603 by loveineq last updated on 26/Jun/21

Let a, b, c > 0 and (a + b) (b + c) = 4 . Prove that

(2 a + b) (a + b) + (b + 2 c) (b + c) ⩾ 8 + \frac{1}{2} (a + 2 b + c) (c + a)

Determine when equality holds .

Answered by mindispower last updated on 27/Jun/21

(2 a + b) (a + b) + (b + 2 c) (b + c)

= {(a + b)}^{2} + {(b + c)}^{2} + a^{2} + a b + b c + c^{2}

{(a + b)}^{2} + {(b + c)}^{2} ⩾ 2 (a + b) (b + c) = 8

a^{2} + c^{2} + a b + b c = b (a + c) + a^{2} + c^{2} . . E

a^{2} + c^{2} ⩾ \frac{{(a + c)}^{2}}{2}

E \Leftrightarrow a^{2} + c^{2} + a b + b c ⩾ \frac{{(a + c)}^{2}}{2} + b (a + c) = (a + c) (b + \frac{a + c}{2})

= \frac{1}{2} (a + c) (2 b + a + c)

\Rightarrow⩾ \frac{1}{2} (a + c) (2 b + a + c) + 8

Commented by loveineq last updated on 27/Jun/21

t h a n k s

Commented by mindispower last updated on 27/Jun/21

p l e a s u r