let-a-b-c-be-positive-real-numbers-such-that-ab-bc-ac-3-prove-the-inquality-a-b-2-c-2-a-2-bc-b-c-2-a-2-b-2-ac-c-b-2-a-2-c-2-ab-3-

Question Number 98983 by M±th+et+s last updated on 17/Jun/20

l e t a, b, c b e p o s i t i v e r e a l n u m b e r s s u c h

t h a t a b + b c + a c = 3

p r o v e t h e i n q u a l i t y

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

Commented by MJS last updated on 17/Jun/20

due to symmetry extremes at a = b = c \Rightarrow

a b + b c + a c = 3 \Leftrightarrow 3 a^{2} = 3 \Rightarrow a = \pm 1

but a > 0 \Rightarrow a = b = c = 1

the inequation with a = b = c turns into

3 a ⩾ 3 \Rightarrow true for a = 1

now test if this is \min or \max by putting

a = . 999; b = 1.001 \Rightarrow c = 1.0000005

\Rightarrow lhs > 3

\Rightarrow proven

I know you want a different kind of proof

but this is the easiest path

Commented by M±th+et+s last updated on 17/Jun/20

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