Find \( f(\alpha) \) and \( f(\beta) \)
If
\[ f(x)=27x^3+\frac1{x^3} \]
and \(\alpha,\beta\) are roots of
\[ 3x+\frac1x=12, \]
then
(a) \(f(\alpha)\ne f(\beta)\)
(b) \(f(\alpha)=10\)
(c) \(f(\beta)=-10\)
(d) none of these
Given,
\[ 3x+\frac1x=12 \]
Cubing both sides,
\[ \left(3x+\frac1x\right)^3=12^3 \]
\[ 27x^3+\frac1{x^3} +9\left(3x+\frac1x\right) =1728 \]
\[ f(x)+9(12)=1728 \]
\[ f(x)=1728-108 \]
\[ f(x)=1620 \]
Therefore,
\[ f(\alpha)=f(\beta)=1620 \]
None of the given options is correct.
\[ \boxed{\text{Correct Answer: (d)}} \]