find the least value

b,a,c are three positive numbers in G.P. and a+b+c=abc, then least value of a^4+a^2+7 is equal to

(where a^4 is a to the power 4 and a^2 is a to the power 2)

2 Answers

1357

Manish Shankar ·2009-07-25 07:57:36

try AMâ‰¥GM

this might help

106

Asish Mahapatra ·2009-07-26 04:26:25

a²=bc
so abc = a(bc) = a³

Now (a+b+c)/3 â‰¥ ³√abc [AM-GM inequality]
==> abc/3 â‰¥ ³√abc [a+b+c=abc given]
==> (abc)^2/3 â‰¥ 3
==> a² â‰¥3 [abc=a³]

So min value of a⁴+a²+7 = 9+3+7 = 19

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