Find \( f(\alpha) \) and \( f(\beta) \)
If
\[ f(x)=64x^3+\frac1{x^3} \]
and \(\alpha,\beta\) are the roots of
\[ 4x+\frac1x=3, \]
then
(a) \(f(\alpha)=f(\beta)=-9\)
(b) \(f(\alpha)=f(\beta)=63\)
(c) \(f(\alpha)\ne f(\beta)\)
(d) none of these
Given,
\[ 4x+\frac1x=3 \]
Cubing both sides,
\[ \left(4x+\frac1x\right)^3=3^3 \]
\[ 64x^3+\frac1{x^3} +12\left(4x\right)\left(\frac1x\right)\left(4x+\frac1x\right) =27 \]
\[ f(x)+48\left(4x+\frac1x\right)=27 \]
\[ f(x)+48(3)=27 \]
\[ f(x)=27-144 \]
\[ f(x)=-117 \]
Therefore,
\[ f(\alpha)=f(\beta)=-117 \]
None of the given options is correct.
\[ \boxed{\text{Correct Answer: (d)}} \]