Question

The function $f(x) = \ln(x+\sqrt{x^2+5})$ is increasing for each x in the interval (where $R$ is the set of all real numbers)

• A. (-\infty, \infty)
• B. (0, \infty)
• C. (-\infty, 0)
• D. $R - (-1, 1)$

Answer 1

Answer: (A), (B), (C)

A, B, C

Answer 2

The derivative of the function $f(x) = \ln(x+\sqrt{x^2+5})$ is $f'(x) = \frac{1}{\sqrt{x^2+5}}$. This derivative is positive for all real numbers $x$. The domain of the function is also all real numbers. Therefore, the function is increasing on $(-\infty, \infty)$. Consequently, it is also increasing on any sub-interval, such as $(0, \infty)$ and $(-\infty, 0)$.