Compare \( f(xy) \) and \( f(x)f(y) \)
Let
\[ f(x)=\sqrt{x^2+1} \]
Then which of the following is correct?
(a) \(f(xy)=f(x)f(y)\)
(b) \(f(xy)\ge f(x)f(y)\)
(c) \(f(xy)\le f(x)f(y)\)
(d) none of these
\[ f(xy)=\sqrt{x^2y^2+1} \]
and
\[ f(x)f(y) = \sqrt{x^2+1}\sqrt{y^2+1} \]
Squaring both sides,
\[ [f(x)f(y)]^2 = (x^2+1)(y^2+1) \]
\[ = x^2y^2+x^2+y^2+1 \]
Since
\[ x^2+y^2\ge0 \]
\[ x^2y^2+1 \le x^2y^2+x^2+y^2+1 \]
Therefore,
\[ f(xy)\le f(x)f(y) \]
\[ \boxed{\text{Correct Answer: (c)}} \]