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The minimum value of `((a^2 +3a+1)(b^2+3b + 1)(c^2 3c+ 1))/(abc)`The minimum value of , where `a, b, c in R` isA. `(11^(3))/(2^(3))`B. 125C. 25D. 27 |
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Answer» Correct Answer - B Let `A=((a^(2)+3a+1)(b^(2)+3b+1)(c^(2)+3c+1))/(abc)` `=((a^(2)+3a+1)/(a))((b^(2)+3b+1)/(b))((c^(2)+3c+1)/(c))` `=(a+3+(1)/(a))(b+3+(1)/(b))(c+3+(1)/(c))` where `a,b,c in R^(+)`. Applying `Am ge GM` on a and `(1)/(a)` `a+(1)/(a)ge 2 " " implies a+(1)/(b)+3ge 5` Similarly, `b+(1)/(b)ge 2 " " implies b+(1)/(b)+3ge 5` and `c+(1)/(c)ge 2 " " implies c+(1)/(c)+3ge 5` `:.(a+(1)/(a)+3)(b+(1)/(b)+3)(c+(1)/(c)+3)ge 125` So, `Age 5*5*5 " " implies Age 125` minimum value of A is 125. |
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