Analysis-Aufstellen von Funktionsgleichungen-Ganzrationale Funktion


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Beispiel Nr: 06
$ \\ \text{ Aufstellen von Funktionsgleichungen}\\ \\ \textbf{Aufgabe:}\\eine ganzrationale Funktion 3.Grades, mit den Nullstellen bei $x_1=-1$ und $x_2=4$ und dem Hochpunkt $H(2/2)$. \\ \\ \textbf{Rechnung:}\\ \\ \text{Funktion} \\ f\left(x\right)=a\cdot x^3+b\cdot x^2+c\cdot x+d\\ f'\left(x\right)=3a\cdot x^2+2b\cdot x+c\\ f''\left(x\right)=6a\cdot x+2b\\ \text{Gegeben:}\\ f\left(-1\right)=0 \qquad a\cdot (-1)^3+b\cdot (-1)^2+c\cdot (-1)+d=0 \\ f\left(4\right)=0 \qquad a\cdot 4^3+b\cdot 4^2+c\cdot 4+d=0 \\ f\left(2\right)=2 \qquad a\cdot 2^3+b\cdot 2^2+c\cdot 2+d=2 \\ f'\left(2\right)=0 \qquad 3a\cdot 2^2+2b\cdot 2+c=0 \\\small \begin{array}{l} -a+b -c+d=0 \\ 64a+16b+4c+d=0 \\ 8a+4b+2c+d=2 \\ 12a+4b+c=0 \\ \\ \end{array} \qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 1 & -1 & 1 & 0 \\ 64 & 16 & 4 & 1 & 0 \\ 8 & 4 & 2 & 1 & 2 \\ 12 & 4 & 1 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{64}{-1}\\z2s1=64-(-1)\cdot \frac{64}{-1}=0 \\ z2s2=16-1\cdot \frac{64}{-1}=80 \\ z2s3=4-(-1)\cdot \frac{64}{-1}=-60 \\ z2s4=1-1\cdot \frac{64}{-1}=65 \\ z2s5=0-0\cdot \frac{64}{-1}=0 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 1 & -1 & 1 & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 8 & 4 & 2 & 1 & 2 \\ 12 & 4 & 1 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{8}{-1}\\z3s1=8-(-1)\cdot \frac{8}{-1}=0 \\ z3s2=4-1\cdot \frac{8}{-1}=12 \\ z3s3=2-(-1)\cdot \frac{8}{-1}=-6 \\ z3s4=1-1\cdot \frac{8}{-1}=9 \\ z3s5=2-0\cdot \frac{8}{-1}=2 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 1 & -1 & 1 & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 12 & -6 & 9 & 2 \\ 12 & 4 & 1 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}1\cdot \frac{12}{-1}\\z4s1=12-(-1)\cdot \frac{12}{-1}=0 \\ z4s2=4-1\cdot \frac{12}{-1}=16 \\ z4s3=1-(-1)\cdot \frac{12}{-1}=-11 \\ z4s4=0-1\cdot \frac{12}{-1}=12 \\ z4s5=0-0\cdot \frac{12}{-1}=0 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 1 & -1 & 1 & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 12 & -6 & 9 & 2 \\ 0 & 16 & -11 & 12 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{1}{80}\\z1s2=1-80\cdot \frac{1}{80}=0 \\ z1s3=-1-(-60)\cdot \frac{1}{80}=-\frac{1}{4} \\ z1s4=1-65\cdot \frac{1}{80}=\frac{3}{16} \\ z1s5=0-0\cdot \frac{1}{80}=0 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & -\frac{1}{4} & \frac{3}{16} & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 12 & -6 & 9 & 2 \\ 0 & 16 & -11 & 12 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{12}{80}\\z3s2=12-80\cdot \frac{12}{80}=0 \\ z3s3=-6-(-60)\cdot \frac{12}{80}=3 \\ z3s4=9-65\cdot \frac{12}{80}=-\frac{3}{4} \\ z3s5=2-0\cdot \frac{12}{80}=2 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & -\frac{1}{4} & \frac{3}{16} & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 16 & -11 & 12 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}2\cdot \frac{16}{80}\\z4s2=16-80\cdot \frac{16}{80}=0 \\ z4s3=-11-(-60)\cdot \frac{16}{80}=1 \\ z4s4=12-65\cdot \frac{16}{80}=-1 \\ z4s5=0-0\cdot \frac{16}{80}=0 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & -\frac{1}{4} & \frac{3}{16} & 0 \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 1 & -1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-\frac{1}{4}}{3}\\z1s3=-\frac{1}{4}-3\cdot \frac{-\frac{1}{4}}{3}=0 \\ z1s4=\frac{3}{16}-(-\frac{3}{4})\cdot \frac{-\frac{1}{4}}{3}=\frac{1}{8} \\ z1s5=0-2\cdot \frac{-\frac{1}{4}}{3}=\frac{1}{6} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & \frac{1}{8} & \frac{1}{6} \\ 0 & 80 & -60 & 65 & 0 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 1 & -1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-60}{3}\\z2s3=-60-3\cdot \frac{-60}{3}=0 \\ z2s4=65-(-\frac{3}{4})\cdot \frac{-60}{3}=50 \\ z2s5=0-2\cdot \frac{-60}{3}=40 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & \frac{1}{8} & \frac{1}{6} \\ 0 & 80 & 0 & 50 & 40 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 1 & -1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}3\cdot \frac{1}{3}\\z4s3=1-3\cdot \frac{1}{3}=0 \\ z4s4=-1-(-\frac{3}{4})\cdot \frac{1}{3}=-\frac{3}{4} \\ z4s5=0-2\cdot \frac{1}{3}=-\frac{2}{3} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & \frac{1}{8} & \frac{1}{6} \\ 0 & 80 & 0 & 50 & 40 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 0 & -\frac{3}{4} & -\frac{2}{3} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}4\cdot \frac{\frac{1}{8}}{-\frac{3}{4}}\\z1s4=\frac{1}{8}-(-\frac{3}{4})\cdot \frac{\frac{1}{8}}{-\frac{3}{4}}=0 \\ z1s5=\frac{1}{6}-(-\frac{2}{3})\cdot \frac{\frac{1}{8}}{-\frac{3}{4}}=\frac{1}{18} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & 0 & \frac{1}{18} \\ 0 & 80 & 0 & 50 & 40 \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 0 & -\frac{3}{4} & -\frac{2}{3} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}4\cdot \frac{50}{-\frac{3}{4}}\\z2s4=50-(-\frac{3}{4})\cdot \frac{50}{-\frac{3}{4}}=0 \\ z2s5=40-(-\frac{2}{3})\cdot \frac{50}{-\frac{3}{4}}=-4\frac{4}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & 0 & \frac{1}{18} \\ 0 & 80 & 0 & 0 & -4\frac{4}{9} \\ 0 & 0 & 3 & -\frac{3}{4} & 2 \\ 0 & 0 & 0 & -\frac{3}{4} & -\frac{2}{3} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}4\cdot \frac{-\frac{3}{4}}{-\frac{3}{4}}\\z3s4=-\frac{3}{4}-(-\frac{3}{4})\cdot \frac{-\frac{3}{4}}{-\frac{3}{4}}=0 \\ z3s5=2-(-\frac{2}{3})\cdot \frac{-\frac{3}{4}}{-\frac{3}{4}}=2\frac{2}{3} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline-1 & 0 & 0 & 0 & \frac{1}{18} \\ 0 & 80 & 0 & 0 & -4\frac{4}{9} \\ 0 & 0 & 3 & 0 & 2\frac{2}{3} \\ 0 & 0 & 0 & -\frac{3}{4} & -\frac{2}{3} \\ \end{array} \\ \\ a=\frac{\frac{1}{18}}{-1}=-\frac{1}{18}\\b=\frac{-4\frac{4}{9}}{80}=-\frac{1}{18}\\c=\frac{2\frac{2}{3}}{3}=\frac{8}{9}\\d=\frac{-\frac{2}{3}}{-\frac{3}{4}}=\frac{8}{9}\\L=\{-\frac{1}{18}/-\frac{1}{18}/\frac{8}{9}/\frac{8}{9}\} \\ \text{Funktion} \\ f\left(x\right)=-\frac{1}{18}x^3-\frac{1}{18}x^2+\frac{8}{9}x+\frac{8}{9} $