Let $f$ and $g$ be the functions defined on the set $\mathbb { R }$ of real numbers by

$$f ( x ) = \mathrm { e } ^ { x } \quad \text { and } \quad g ( x ) = \mathrm { e } ^ { - x } .$$

We denote by $\mathscr { C } _ { f }$ the representative curve of function $f$ and $\mathscr { C } _ { g }$ that of function $g$ in an orthonormal coordinate system of the plane.

For every real number $a$, we denote by $M$ the point of $\mathscr { C } _ { f }$ with abscissa $a$ and $N$ the point of $\mathscr { C } _ { g }$ with abscissa $a$.

The tangent line to $\mathscr { C } _ { f }$ at $M$ intersects the $x$-axis at $P$, the tangent line to $\mathscr { C } _ { g }$ at $N$ intersects the $x$-axis at $Q$.

Questions 1 and 2 can be treated independently.

\begin{enumerate}
  \item Prove that the tangent line to $\mathscr { C } _ { f }$ at $M$ is perpendicular to the tangent line to $\mathscr { C } _ { g }$ at $N$.
  \item a. What can be conjectured about the length $PQ$?\\
b. Prove this conjecture.
\end{enumerate}