IIT JEE 1983 - 'Adapted' - Subjective to Multi-Correct Q8

Let $f(x)=1+x\ln\left(x+\sqrt{x^2+1}\right)$ and $g(x)=\sqrt{1+x^2}$ . Then which of the following is/are correct?

• A: $f(x)>g(x) \ \forall \ x \in \mathbb R^+$
• B: $f(x)<g(x) \ \forall \ x \in \mathbb R^-$
• C: There exists $x=a>0$ for which $f(x)<g(x)$
• D: There exists $x=a<0$ for which $f(x)>g(x)$

Enter your answer as a 4 digit string of 1s and 9s, using 1 for correct option, 9 for wrong. For example, 1199 indicates A and B are correct, C and D are incorrect. None, one or all may also be correct.

In case you are preparing for IIT JEE, you may want to try IIT JEE 1983 Mathematics Archives .