Berechne $f^{\mathrm{\prime }}(x)f\text{'}(x)$

$f^{\mathrm{\prime }}(x)=-x+3f\text{'}(x)=-x+3$

Setze mit der Steigung $mm$ gleich.

$-2=-x+3-2=-x+3$

Löse nach $xx$ bzw. $x_{0}x\_0$ auf.

$x_0=5x\_0=5$

Berechne $f(x_0)f(x\_0)$

$f(5)=-{\displaystyle \frac{25}{2}}+15={\displaystyle \frac{5}{2}}f(5)=-\backslash dfrac\{25\}\{2\}+15=\backslash dfrac\{5\}\{2\}$

Setze $x_0,f(x_0),mx\_0,f(x\_0),m$ in die Tangentenformel ein.

$g(x)=f^{\mathrm{\prime }}(x_0)(x-x_0)+f(x_0)g(x)=f\text{'}(x\_0)(x-x\_0)+f(x\_0)$ $\backslash \backslash$ $\phantom{g(x)}=-2(x-5)+{\displaystyle \frac{5}{2}}\backslash phantom\{g(x)\}=-2(x-5)+\backslash dfrac\{5\}\{2\}$

Berechne $f^{\mathrm{\prime }}(x)f\text{'}(x)$

$f^{\mathrm{\prime }}(x)=-x^2+x+11f\text{'}(x)=-x^2+x+11$

Setze mit der Steigung $mm$ gleich.

$5=-x^2+x+115=-x^2+x+11$

Zwischenschritt: Bringe in die Nullform $\backslash \backslash$ und bestimme $p,qp,q$.

$0=x^2-x-60=x^2-x-6$

Berechne $f(x_1),f(x_2)f(x\_1),f(x\_2)$

$f(3)=-{\displaystyle \frac{1}{3}}\cdot 3^3+{\displaystyle \frac{1}{2}}\cdot 3^2+11\cdot 3-6f(3)=-\backslash dfrac\{1\}\{3\}\backslash cdot\; 3^3+\backslash dfrac\{1\}\{2\}\backslash cdot\; 3^2+11\backslash cdot\; 3-6$ $\backslash \backslash$ $\phantom{f(3)}=-9+{\displaystyle \frac{9}{2}}+33-6={\displaystyle \frac{45}{2}}\backslash phantom\{f(3)\}=-9+\backslash dfrac\{9\}\{2\}+33-6=\backslash dfrac\{45\}\{2\}$ $\backslash \backslash$ $f(-2)=-{\displaystyle \frac{1}{3}}\cdot (-2{)}^3+{\displaystyle \frac{1}{2}}\cdot (-2{)}^2+11\cdot (-2)-6f(-2)=-\backslash dfrac\{1\}\{3\}\backslash cdot\; (-2)^3+\backslash dfrac\{1\}\{2\}\backslash cdot\; (-2)^2+11\backslash cdot\; (-2)-6$ $\backslash \backslash$ $\phantom{f(-2)}={\displaystyle \frac{8}{3}}+2-22-6=-{\displaystyle \frac{70}{3}}\backslash phantom\{f(-2)\}=\backslash dfrac\{8\}\{3\}+2-22-6=-\backslash dfrac\{70\}\{3\}$

Setze $x_1,f(x_1),mx\_1,f(x\_1),m$ in die $\backslash \backslash$ Tangentenformel ein.

$g_1(x)=f^{\mathrm{\prime }}(x_0)(x-x_0)+f(x_0)g\_1(x)=f\text{'}(x\_0)(x-x\_0)+f(x\_0)$ $\backslash \backslash$ $\phantom{g(x)}=5(x-3)+{\displaystyle \frac{45}{2}}\backslash phantom\{g(x)\}=5(x-3)+\backslash dfrac\{45\}\{2\}$

Setze $x_2,f(x_2),mx\_2,f(x\_2),m$ in die $\backslash \backslash$ Tangentenformel ein.

$g_2(x)=f^{\mathrm{\prime }}(x_0)(x-x_0)+f(x_0)g\_2(x)=f\text{'}(x\_0)(x-x\_0)+f(x\_0)$ $\backslash \backslash$ $\phantom{g(x)}=5(x+2)-{\displaystyle \frac{70}{3}}\backslash phantom\{g(x)\}=5(x+2)-\backslash dfrac\{70\}\{3\}$