If a- b- c-0- a-b-c-6 and fx-6-x-1- find the minimum value of fa - fb - fc -

The correct option is C 8f(a).f(b).f(c) = (6 a)(6 b)(6 c)/abc = 216 36(a+b+c)+6(ab + bc + ca) abc/abc = (ab + bc + ca)/abc 1 = 6(1/a +1/b + 1/c) 1 As A ...

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Quantitative Aptitude

A.M Greater than or Equal to G.M

If a, b, c>0,...

Question

If a, b, c >0, a + b + c =6 and f(x) = $(\frac{6}{x})$ - 1, find the minimum value of f(a).f(b).f(c)?

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Solution

The correct option is D
8

f(a).f(b).f(c) = $\frac{(6 - a) (6 - b) (6 - c)}{a b c}$ = $\frac{216 - 36 (a + b + c) + 6 (a b + b c + c a) - a b c}{a b c}$ = $\frac{(a b + b c + c a)}{a b c}$ - 1 = 6 $(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ - 1
As AM $\geq$ HM
$\frac{(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})}{3}$ = $\frac{\frac{6}{a} + \frac{6}{b} + \frac{6}{c}}{3}$ > 3
$\frac{6}{a} + \frac{6}{b} + \frac{6}{c}$ > 9
f(a).f(b).f(c)> 9 - 1 = 8

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If a, b, c >0, a + b + c =6 and f(x) = (6x) - 1, find the minimum value of f(a).f(b).f(c)?

The correct option is D 8f(a).f(b).f(c) = (6−a)(6−b)(6−c)abc = 216−36(a+b+c)+6(ab+bc+ca)−abcabc = (ab+bc+ca)abc - 1 = 6(1a+1b+1c) - 1 As AM≥HM (1a+1b+1c)3 = 6a+6b+6c3 > 3 6a+6b+6c > 9 f(a).f(b).f(c)> 9 - 1 = 8

If a, b, c >0, a + b + c =6 and f(x) = $(\frac{6}{x})$ - 1, find the minimum value of f(a).f(b).f(c)?