ex. 2

analyse_1_td_1.tex

@@ -2,6 +2,7 @@
 
 \usepackage{mathtools}
 \usepackage{amssymb}
+\usepackage{pgfplots}
 
 \title{Analyse 1 - TD 1 \\ NON CORRIGÉ}
 \author{Timéo Pochin}
@@ -92,4 +93,83 @@
 
     \subsection*{k)}
 
+    \section*{Exercice 2}
+
+    \[
+    f:x\mapsto ax+b
+    \]
+
+    \begin{align}
+                  & \big(|f(-1)|=3\big)\land\big(|f(2)=2|\big) \nonumber \\
+        \iff\quad & (|b-a|=3)\land(|2a+b|=2) \nonumber \\
+        \iff\quad & (b-a=3\lor b-a=-3)\land(2a+b=2\lor 2a+b=-2) \nonumber \\
+        \iff\quad & (b-a=3\land 2a+b=2) \nonumber \\
+             \lor & (b-a=3\land 2a+b=-2) \nonumber \\
+             \lor & (b-a=-3\land 2a+b=2) \nonumber \\
+             \lor & (b-a=-3\land 2a+b=-2) \nonumber
+    \end{align}
+
+    \begin{align}
+                  & b-a=3\land 2a+b=2 \nonumber \\
+        \iff\quad & 3a=-1\land 3b=8 \nonumber \\
+        \iff\quad & a=-\frac{1}{3}\land b=\frac{8}{3} \nonumber \\
+    \end{align}
+    \begin{align}
+                  & b-a=3\land 2a+b=-2 \nonumber \\
+        \iff\quad & 3a=-5\land 3b=4 \nonumber \\
+        \iff\quad & a=-\frac{5}{3}\land b=\frac{4}{3} \nonumber \\
+    \end{align}
+    \begin{align}
+                  & b-a=-3\land 2a+b=2 \nonumber \\
+        \iff\quad & 3a=5\land 3b=-4 \nonumber \\
+        \iff\quad & a=\frac{5}{3}\land b=-\frac{4}{3} \nonumber \\
+    \end{align}
+    \begin{align}
+                  & b-a=-3\land 2a+b=-2 \nonumber \\
+        \iff\quad & 3a=1\land 3b=-8 \nonumber \\
+        \iff\quad & a=\frac{1}{3}\land b=-\frac{8}{3} \nonumber \\
+    \end{align}
+
+    Donc
+
+    \[
+    (a,b)\in
+    \left\{
+        \left(-\frac{1}{3},\frac{8}{3}\right),
+        \left(-\frac{5}{3},\frac{4}{3}\right),
+        \left(\frac{5}{3},-\frac{4}{3}\right),
+        \left(\frac{1}{3},-\frac{8}{3}\right)
+    \right\}
+    \]
+
+    \begin{tikzpicture}
+        \begin{axis}[
+            xmin = -5, xmax = 5,
+            ymin = -5, ymax = 5,
+            axis x line = middle,
+            axis y line = middle,
+            xtick distance = 1,
+            ytick distance = 1,
+            grid = both,
+            minor tick num = 5,
+            major grid style = {lightgray},
+            minor grid style = {lightgray!25},
+            width = \textwidth,
+            height = \textwidth
+        ]
+        \addplot[
+            color = black
+        ]{-(1/3)*x+(8/3)};
+        \addplot[
+            color = black
+        ]{-(5/3)*x+(4/3)};
+        \addplot[
+            color = black
+        ]{(1/3)*x-(8/3)};
+        \addplot[
+            color = black
+        ]{(5/3)*x-(4/3)};
+        \end{axis}
+    \end{tikzpicture}
+
 \end{document}