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Version 6.3 von Dirk Tebbe am 2025/10/13 14:14
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1 1.
2 [[image:Exponentialfunktion.svg||width="450" style="display:block;margin-left:auto;margin-right:auto"]]
3 1.
4 {{formula}}f\left(x\right)=0{{/formula}}
5 {{formula}}4-\frac{1}{2} e^x=0{{/formula}}
6 {{formula}}4=\frac{1}{2} e^x{{/formula}}
7 {{formula}}8=e^x{{/formula}}
8 {{formula}}ln(8)=x{{/formula}}
9
10 {{formula}}f´\left(x\right)=-\frac{1}{2} e^x{{/formula}}
11 {{formula}}f´\left(ln(8)\right)=-\frac{1}{2} e^{ln(8)}=-\frac{1}{2}\cdot 8=-4{{/formula}}
12
13 Einsetzen von {{formula}}m=-4{{/formula}} in {{formula}}y=mx+c{{/formula}}:
14 {{formula}}y= -4x+c{{/formula}}
15 und {{formula}}N(ln(8)|0){{/formula}}
16 {{formula}} 0= -4 \cdot ln(8)+c{{/formula}}
17 {{formula}} c = 4 \cdot ln(8){{/formula}}
18
19 {{formula}}y=-4\cdot x+ 4 \cdot ln(8){{/formula}}
20
21 {{formula}}4-\frac{1}{2} e^x=4{{/formula}}
22 {{formula}}-\frac{1}{2} e^x=0{{/formula}}
23 {{formula}} e^x=0{{/formula}}
24 Diese Gleichung hat keine Lösung, da {{formula}} e^x\neq 0{{/formula}}
25
26 {{formula}}f´\left(x\right)=-\frac{1}{2} e^x< 0{{/formula}} für alle x.