Let $f$ be the differentiable function defined on the interval $] 0 ; + \infty [$ by

$$f ( x ) = \mathrm { e } ^ { x } + \frac { 1 } { x }$$

\textbf{1. Study of an auxiliary function}

a. Let the function $g$ be differentiable, defined on $[ 0 ; + \infty [$ by

$$g ( x ) = x ^ { 2 } \mathrm { e } ^ { x } - 1 .$$

Study the direction of variation of the function $g$.

b. Prove that there exists a unique real number $a$ belonging to $[ 0 ; + \infty [$ such that $g ( a ) = 0$.

Prove that $a$ belongs to the interval $[ 0{,}703 ; 0{,}704 [$.

c. Determine the sign of $g ( x )$ on $[ 0 ; + \infty [$.

\textbf{2. Study of the function $f$}

a. Determine the limits of the function $f$ at 0 and at $+ \infty$.

b. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $] 0 ; + \infty [$.

Prove that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.

c. Deduce the direction of variation of the function $f$ and draw its variation table on the interval $] 0 ; + \infty [$.

d. Prove that the function $f$ has as its minimum the real number

$$m = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a } .$$

e. Justify that $3{,}43 < m < 3{,}45$.