In this part, we consider that the function $f$, defined and twice differentiable on $[0; +\infty[$, is defined by $$f(x) = (4x - 2)\mathrm{e}^{-x + 1}.$$ We will denote respectively $f'$ and $f''$ the derivative and second derivative of the function $f$.
Study of the function $f$ a. Show that $f'(x) = (-4x + 6)\mathrm{e}^{-x + 1}$. b. Use this result to determine the complete table of variations of the function $f$ on $[0; +\infty[$. It is admitted that $\lim_{x \rightarrow +\infty} f(x) = 0$. c. Study the convexity of the function $f$ and specify the abscissa of any possible inflection point of the representative curve of $f$.
We consider a function $F$ defined on $[0; +\infty[$ by $F(x) = (ax + b)\mathrm{e}^{-x + 1}$, where $a$ and $b$ are two real numbers. a. Determine the values of the real numbers $a$ and $b$ such that the function $F$ is a primitive of the function $f$ on $[0; +\infty[$. b. It is admitted that $F(x) = (-4x - 2)\mathrm{e}^{-x + 1}$ is a primitive of the function $f$ on $[0; +\infty[$. Deduce the exact value, then an approximate value to $10^{-2}$ near, of the integral $$I = \int_{\frac{3}{2}}^{8} f(x)\mathrm{d}x.$$
A municipality has decided to build a freestyle scooter track. The profile of this track is given by the representative curve of the function $f$ on the interval $[\frac{3}{2}; 8]$. The unit of length is the meter. a. Give an approximate value to the nearest cm of the height of the starting point D. b. The municipality has organized a graffiti competition to decorate the wall profile of the track. The selected artist plans to cover approximately $75\%$ of the wall surface. Knowing that a 150 mL aerosol can covers a surface of $0.8\mathrm{~m}^2$, determine the number of cans she will need to use to create this work.
In this part, we consider that the function $f$, defined and twice differentiable on $[0; +\infty[$, is defined by
$$f(x) = (4x - 2)\mathrm{e}^{-x + 1}.$$
We will denote respectively $f'$ and $f''$ the derivative and second derivative of the function $f$.
\begin{enumerate}
\item Study of the function $f$\\
a. Show that $f'(x) = (-4x + 6)\mathrm{e}^{-x + 1}$.\\
b. Use this result to determine the complete table of variations of the function $f$ on $[0; +\infty[$. It is admitted that $\lim_{x \rightarrow +\infty} f(x) = 0$.\\
c. Study the convexity of the function $f$ and specify the abscissa of any possible inflection point of the representative curve of $f$.
\item We consider a function $F$ defined on $[0; +\infty[$ by $F(x) = (ax + b)\mathrm{e}^{-x + 1}$, where $a$ and $b$ are two real numbers.\\
a. Determine the values of the real numbers $a$ and $b$ such that the function $F$ is a primitive of the function $f$ on $[0; +\infty[$.\\
b. It is admitted that $F(x) = (-4x - 2)\mathrm{e}^{-x + 1}$ is a primitive of the function $f$ on $[0; +\infty[$. Deduce the exact value, then an approximate value to $10^{-2}$ near, of the integral
$$I = \int_{\frac{3}{2}}^{8} f(x)\mathrm{d}x.$$
\item A municipality has decided to build a freestyle scooter track. The profile of this track is given by the representative curve of the function $f$ on the interval $[\frac{3}{2}; 8]$. The unit of length is the meter.\\
a. Give an approximate value to the nearest cm of the height of the starting point D.\\
b. The municipality has organized a graffiti competition to decorate the wall profile of the track. The selected artist plans to cover approximately $75\%$ of the wall surface. Knowing that a 150 mL aerosol can covers a surface of $0.8\mathrm{~m}^2$, determine the number of cans she will need to use to create this work.
\end{enumerate}