We consider the function $f$ defined on the interval $]2; +\infty[$ by $$f(x) = x\ln(x-2)$$ Part of the representative curve $\mathscr{C}_f$ of the function $f$ is given below.
Conjecture, using the graph, the direction of variation of $f$, its limits at the boundaries of its domain of definition, and any possible asymptotes.
Solve the equation $f(x) = 0$ on $]2; +\infty[$.
Calculate $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} f(x)$. Does this result confirm one of the conjectures made in question 1?
Prove that for all $x$ belonging to $]2; +\infty[$: $$f'(x) = \ln(x-2) + \frac{x}{x-2}$$
We consider the function $g$ defined on the interval $]2; +\infty[$ by $g(x) = f'(x)$. a. Prove that for all $x$ belonging to $]2; +\infty[$, we have: $$g'(x) = \frac{x-4}{(x-2)^2}$$ b. We admit that $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} g(x) = +\infty$ and that $\displaystyle\lim_{x \rightarrow +\infty} g(x) = +\infty$. Deduce the table of variations of the function $g$ on $]2; +\infty[$. The exact value of the extremum of the function $g$ should be shown. c. Deduce that, for all $x$ belonging to $]2; +\infty[$, $g(x) > 0$. d. Deduce the direction of variation of the function $f$ on $]2; +\infty[$.
Study the convexity of the function $f$ on $]2; +\infty[$ and specify the coordinates of any possible inflection point of the representative curve of the function $f$.
How many values of $x$ exist for which the representative curve of $f$ admits a tangent with slope equal to 3?
We consider the function $f$ defined on the interval $]2; +\infty[$ by
$$f(x) = x\ln(x-2)$$
Part of the representative curve $\mathscr{C}_f$ of the function $f$ is given below.
\begin{enumerate}
\item Conjecture, using the graph, the direction of variation of $f$, its limits at the boundaries of its domain of definition, and any possible asymptotes.
\item Solve the equation $f(x) = 0$ on $]2; +\infty[$.
\item Calculate $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} f(x)$.\\
Does this result confirm one of the conjectures made in question 1?
\item Prove that for all $x$ belonging to $]2; +\infty[$:
$$f'(x) = \ln(x-2) + \frac{x}{x-2}$$
\item We consider the function $g$ defined on the interval $]2; +\infty[$ by $g(x) = f'(x)$.\\
a. Prove that for all $x$ belonging to $]2; +\infty[$, we have:
$$g'(x) = \frac{x-4}{(x-2)^2}$$
b. We admit that $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} g(x) = +\infty$ and that $\displaystyle\lim_{x \rightarrow +\infty} g(x) = +\infty$.\\
Deduce the table of variations of the function $g$ on $]2; +\infty[$. The exact value of the extremum of the function $g$ should be shown.\\
c. Deduce that, for all $x$ belonging to $]2; +\infty[$, $g(x) > 0$.\\
d. Deduce the direction of variation of the function $f$ on $]2; +\infty[$.
\item Study the convexity of the function $f$ on $]2; +\infty[$ and specify the coordinates of any possible inflection point of the representative curve of the function $f$.
\item How many values of $x$ exist for which the representative curve of $f$ admits a tangent with slope equal to 3?
\end{enumerate}