Mathematics
Question
Consider the function f(x) = ln(x^2 + 1).
a) Find the domain of f(x) and determine any points of discontinuity.
b) Calculate the first and second derivatives of f(x).
c) Determine the intervals where the function is increasing or decreasing, and identify any local extrema.
d) Investigate the concavity of the function and locate any inflection points.
Answer to it
Question 1a
Consider the function \( f(x) = \ln(x^2 + 1) \).
a) Find the domain of \( f(x) \) and determine any points of discontinuity.
Solution
a) The natural logarithm function is defined only for positive values. Therefore, \( x^2 + 1 \) must be
greater than 0:
\[ x^2 + 1 > 0 \]
Solving for \( x^2 \), we get \( x^2 > -1 \), which is true for all real numbers \( x \). Thus, the domain of
\( f(x) \) is all real numbers.
Since the domain is the entire real number line, there are no points of discontinuity for \( f(x) \).
Question 1b
Question
Consider the function f(x) = ln(x^2 + 1).
a) Find the domain of f(x) and determine any points of discontinuity.
b) Calculate the first and second derivatives of f(x).
c) Determine the intervals where the function is increasing or decreasing, and identify any local extrema.
d) Investigate the concavity of the function and locate any inflection points.
Answer to it
Question 1a
Consider the function \( f(x) = \ln(x^2 + 1) \).
a) Find the domain of \( f(x) \) and determine any points of discontinuity.
Solution
a) The natural logarithm function is defined only for positive values. Therefore, \( x^2 + 1 \) must be
greater than 0:
\[ x^2 + 1 > 0 \]
Solving for \( x^2 \), we get \( x^2 > -1 \), which is true for all real numbers \( x \). Thus, the domain of
\( f(x) \) is all real numbers.
Since the domain is the entire real number line, there are no points of discontinuity for \( f(x) \).
Question 1b
