We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$. The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system. The purpose of this part is to study the relative position of these two curves.
After observing the graph, what conjecture can be made?
Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
In this question, we consider the interval $]0; +\infty[$. We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$. a. Show that, for all strictly positive real $x$, $$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$ It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$. b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.) c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
a. Is the conjecture made in question 1 of Part B valid? b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$. c. Show that at this point $A$, these two curves have the same tangent line.
We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$.\\
The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system.\\
The purpose of this part is to study the relative position of these two curves.
\begin{enumerate}
\item After observing the graph, what conjecture can be made?
\item Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
\item In this question, we consider the interval $]0; +\infty[$.\\
We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$.\\
a. Show that, for all strictly positive real $x$,
$$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$
It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$.\\
b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.)\\
c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
\item a. Is the conjecture made in question 1 of Part B valid?\\
b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$.\\
c. Show that at this point $A$, these two curves have the same tangent line.
\end{enumerate}