We consider the function $f$ defined on $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x }$$ We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system of the plane. We admit that $f$ is twice differentiable on $[ 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.
- By noting that for all $x$ in $[ 0 ; + \infty [$, we have
$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } }$$ prove that the curve $\mathscr { C } _ { f }$ has an asymptote at $+ \infty$ for which you will give an equation.
2. Prove that for all real $x$ belonging to $[ 0 ; + \infty [$ : $$f ^ { \prime } ( x ) = ( 1 - x ) \mathrm { e } ^ { - x }$$
- Draw up the table of variations of $f$ on $[ 0 ; + \infty [$, on which you will show the values at the boundaries as well as the exact value of the extremum.
- Determine, on the interval $[ 0 ; + \infty [$, the number of solutions of the equation
$$f ( x ) = \frac { 367 } { 1000 }$$
- We admit that for all $x$ belonging to $[ 0 ; + \infty [$ :
$$f ^ { \prime \prime } ( x ) = \mathrm { e } ^ { - x } ( x - 2 )$$ Study the convexity of the function $f$ on the interval $[ 0 ; + \infty [$. 6. Let $a$ be a real number belonging to $[ 0 ; + \infty [$ and A the point of the curve $\mathscr { C } _ { f }$ with abscissa $a$. We denote $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at A. We denote $\mathrm { H } _ { a }$ the point of intersection of the line $T _ { a }$ and the ordinate axis. We denote $g ( a )$ the ordinate of $\mathrm { H } _ { a }$. a. Prove that a reduced equation of the tangent $T _ { a }$ is: $$y = \left[ ( 1 - a ) \mathrm { e } ^ { - a } \right] x + a ^ { 2 } \mathrm { e } ^ { - a }$$ b. Deduce the expression of $g ( a )$. c. Prove that $g ( a )$ is maximum when A is an inflection point of the curve $\mathscr { C } _ { f }$.