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\( \begin{array}{l} f(k)=\frac{x}{a} \cdot e^{a x}=\frac{1}{a} x \cdot e^{a x} \\ 0=\frac{1}{a x}(x) \\ 0=\frac{1}{a} x \cdot e^{a x} / \ln u=\frac{1}{a} x \quad u^{\prime}(x)=\frac{1}{a} \\ f \quad v \quad L^{a x} \quad v \quad v(x)=e^{a x} \quad v^{\prime}(x)=a \cdot e^{a x} \\ \frac{1}{a} x=0 \quad x=0 \end{array} \)
Extramar
\( \begin{array}{l} f_{u}^{\prime}(x)=u^{\prime}(x) \cdot v(x)+u^{2}(x) \cdot v^{\prime}(x) \\ \begin{array}{l} f_{1}^{\prime}(x)=u(x) \cdot v(x)+u(x) \cdot v(x) \\ f^{\prime}(x)=\frac{1}{a} \cdot e^{a x}+\frac{1}{a} x \cdot a \cdot e^{a x}=\frac{1}{a} \cdot e^{a x}+x \cdot e^{a x} \end{array} \\ f_{a}^{\prime}(x)=e^{a^{x}} \cdot\left(\frac{1}{a}+x\right) \\ \frac{1}{a}+x=0 \quad 1-\frac{1}{a} \\ x_{t}=-\frac{1}{a} \\ f_{4}\left(-\frac{1}{a}\right)=\frac{1}{a} \cdot\left(-\frac{1}{a}\right) \cdot e^{\frac{t}{a \cdot\left(-\frac{1}{a}\right)}>-\frac{1}{a^{2}}} \cdot e^{-1} \approx 0,37 \cdot\left(-\frac{1}{a^{2}}\right) \\ \text { E }\left(\left.-\frac{1}{a} \right\rvert\, 0,37 \cdot\left(-\frac{1}{a^{2}}\right)\right. \\ \end{array} \)
\( \sqrt[\xi]{3} \)
\( \begin{array}{l} f_{a}^{\prime}(x)=e^{a x} \cdot\left(\frac{1}{a}+x\right) \quad u(x)=e^{a x} \quad u^{\prime}(x)=a \cdot e^{a x} \\ v(x)=\frac{1}{a}+x \quad v^{\prime}(x)=1 \\ \end{array} \)
\( \begin{array}{l} f_{a}^{\prime \prime}(x)=e^{a x} \cdot 1+a \cdot e^{a x} \cdot\left(\frac{1}{a}+x\right)=e^{a x}+e^{a x}\left(\frac{1}{a}+x\right) \\ f_{a}^{\prime \prime}(x)=e^{a x}\left(1+\frac{1}{a}+x\right) \\ \sqrt[3]{\xi} \\ \left.1+\frac{1}{a}+x=0 \quad \right\rvert\,-1-\frac{1}{a} \\ x_{w}=-1-\frac{1}{a} \\ \end{array} \)
\( \operatorname{lum} \) EP:
\( f_{a}^{\prime \prime}\left(-\frac{1}{a}\right)=e^{-\frac{1}{a} \cdot a}\left(1+\frac{1}{a}+\left(-\frac{1}{a}\right)\right)=e^{-1} \cdot 1=e^{-1}>0 \text { V Tifpunlet } \)
E- Funktionen
Aufgabe:
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Problem/Ansatz:
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