Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 209353 by alcohol last updated on 08/Jul/24

Answered by Berbere last updated on 08/Jul/24

$$f(x+y)=f\left(x\right).f\left(y\right)$$$$f\left(x\right)=f(\frac{x}{2}+\frac{x}{2})={\left(f\left(\frac{x}{2}\right)\right)}^2⩾0$$$$\mathrm{\forall }x\in \mathbb{R}f\left(x\right)⩾0$$$$f(x+y)=f\left(x\right)f\left(y\right)\mathrm{\forall }y\in \mathbb{R}fixex\to f(x+y)estderivable$$$$\Rightarrow f^{\prime }(x+y)=f^{\prime }\left(x\right)f\left(y\right)\Rightarrow f^{\prime }\left(x\right)=f^{\prime }\left(x\right)f\left(0\right)$$$$sif\left(0\right)=0\Rightarrow f^{\prime }\left(x\right)=0\Rightarrow f\left(x\right)=c\Rightarrow c=f\left(0\right)=0\Rightarrow f=0_{\mathbb{R}}$$$$f^{\prime }(y+x_0)=f\left(x_0\right)f^{\prime }\left(y\right)$$$$\Rightarrow f^{\prime }\left(x_0\right)=f\left(x_0\right)f^{\prime }\left(0\right)$$$$\Rightarrow festsolution?\{\begin{array}{l}{y}^{\prime }=f^{\prime }\left(0\right)y\\ y\left(0\right)=1\end{array}$$$$y={ke}^{f^{\prime }\left(0\right)x}$$$$y\left(0\right)=1\Rightarrow k=1$$$$f\left(x\right)=e^{f^{\prime }\left(0\right)x}\Rightarrow f\left(x\right)=e^{ax}$$$$$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com