Study of ln(x²−4)

Problem

Study $f(x) = \ln(x^2-4)$ (domain, asymptotes, symmetries, sign, $f'$, $f''$, graph).

Solution

Domain: $x^2-4>0 \Rightarrow x\in(-\infty,-2)\cup(2,+\infty)$. Symmetry: $f(-x)=f(x)$, an even function.

Sign: $f(x)>0 \iff x^2-4>1 \iff |x|>\sqrt5$; $f=0$ at $x=\pm\sqrt5$.

Asymptotes: $\lim_{x\to 2^+}f = -\infty$ and $\lim_{x\to -2^-}f=-\infty$: vertical asymptotes $x=\pm 2$. As $x\to\pm\infty$, $f\to+\infty$ (no horizontal asymptote).

First derivative: $f'(x)=\dfrac{2x}{x^2-4}$. On the branch $x>2$: $f'>0$ (increasing); on the branch $x<-2$: $f'<0$ (decreasing). No stationary point in the domain.

Second derivative: $f''(x)=\dfrac{2(x^2-4)-2x\cdot 2x}{(x^2-4)^2}=\dfrac{-2x^2-8}{(x^2-4)^2}<0$ on the whole domain: concave downwards on both branches.

Links

Topics: Derivatives
Concepts: Asymptote · Derivative of a composite
Functions: Logarithmic function
Skills: Study a function · Sketch a graph
Exercise type: Study of a function