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extremum_x = f_derv.roots[0] + %- set extremum_y = f(f_derv.roots[0]) + %- set f6 = f(6) + %- set fm6 = f(-6) + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=2]{$ x $/1, Variations de $ f(x) $/2}{-6, $\Var{extremum_x}$, 6} + %- if f[2] > 0 + \tkzTabVar{ +/$\Var{fm6}$, -/$\Var{extremum_y}$, +/$\Var{f6}$} + %- else + \tkzTabVar{ -/$\Var{fm6}$, +/$\Var{extremum_y}$, -/$\Var{f6}$} + %- endif + \end{tikzpicture} + \end{center} + \end{itemize} + \item + %- if f[2] > 0 + La fonction a un minimum. + %- else + La fonction a un maximum. + %- endif + Il vaut $\Var{extremum_y}$ et est atteint en $x = \Var{extremum_x}$. + \end{enumerate} \end{solution} -\begin{exercise}[subtitle={Inéquation et tableaux}] +\begin{exercise}[subtitle={Inéquation et tableaux}, points=2] + Tracer le tableau de signe des fonctions suivantes en le démontrant à l'aide de la résolution d'une inéquation. + \begin{multicols}{2} + \begin{enumerate} + %- set f = rdm.expression("x + {a}", global_config={"min_max":(1, 20)}) + \item $f(x) = \Var{f}$ + %- set g = rdm.expression("{a}x + {b}", global_config={"min_max":(1, 20)}) + \item $g(x) = \Var{g}$ + \end{enumerate} + \end{multicols} \end{exercise} \begin{solution} - + \begin{enumerate} + \item + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + %- set racine = -f[0] + \begin{align*} + f(x) & \geq 0 \\ + \Var{f} & \geq 0 \\ + \Var{f + racine} &\geq \Var{0 + racine} \\ + \end{align*} + Donc $f(x)$ est positif quand $x$ est supérieur à $\Var{racine}$. On en déduit le tableau de signe + + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ f(t) $/1}{, $\Var{racine}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + \item + Pour déterminer les valeurs de $x$ pour lesquelles $g(x)$ est positive, il faut résoudre l'inéquation + + %- set cst = -g[0] + %- set coef = g[1] + %- set racine = cst / coef + \begin{align*} + g(x) & \geq 0 \\ + \Var{g} & \geq 0 \\ + \Var{g + cst} &\geq \Var{0 + cst} \\ + \frac{\Var{g + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\ + x &\geq \Var{racine.simplify()} \\ + \end{align*} + + Donc $f(x)$ est positif quand $x$ est supérieur à $\Var{racine}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ g(t) $/1}{, $\Var{racine}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + \end{enumerate} \end{solution} \end{document}