Question-74213

Question Number 74213 by FCB last updated on 20/Nov/19

Commented by MJS last updated on 20/Nov/19

information missing

I think the question is find minimum of the

expression

this should be at a = b = c = \frac{1}{3}

in this case the answer is \frac{69}{4} = 17.25

Answered by mind is power last updated on 20/Nov/19

w h a t i s q u a t i o n ?

Answered by mind is power last updated on 20/Nov/19

a = b = 0, c = 1

= 14 + \frac{7}{2} = \frac{35}{2} = 17, 5

m i n \frac{7 + 2 b}{1 + a} + \frac{7 + 2 c}{1 + b} + \frac{7 + 2 a}{1 + c}

a + b + c = 1, a, b, c ⩾ 0

x = 1 + a, y = 1 + b, z = 1 + c \Rightarrow x + y + z = 4

\Leftrightarrow f (x, y, z) \frac{2 y + 5}{x} + \frac{5 + 2 z}{y} + \frac{5 + 2 x}{z} = 5 (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) + 2 (\frac{y}{x} + \frac{z}{y} + \frac{x}{z})

w e w i l l u s e H a - G M - A R i n q u a l i t e

\frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} ⩽ \sqrt[3]{x y z} ⩽ \frac{x + y + z}{3} \Rightarrow \frac{1}{x} + \frac{1}{y} + \frac{1}{z} ⩾ \frac{9}{x + y + z} = \frac{9}{4}

(\frac{y}{x} + \frac{z}{y} + \frac{x}{z}) ⩾ 3 \sqrt[3]{(\frac{y . x . z}{x . y . z})} = 3

\Rightarrow f (x, y, z) ⩾ 6 + \frac{9}{4} . 5 = 6 + 11.25 = 17.25

e q u a l i t i y w h e n x = y = z = \frac{4}{3} \Leftrightarrow a = b = c = \frac{1}{3}