Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 158671 by mahdipoor last updated on 07/Nov/21

∀x∈R ; f(x)=f ′(x)  prove f(x+y)=f(x)f(y)

$$\forall{x}\in{R}\:;\:{f}\left({x}\right)={f}\:'\left({x}\right) \\ $$$${prove}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$

Commented by mr W last updated on 07/Nov/21

generally  for f(x+y)=f(x)f(y) we have   f(x)=a^x  (a>0)  f′(x)=(ln a)a^x ≠f(x) (except a=e)  therefore  f′(x)=f(x) doesn′t mean   f(x+y)=f(x)f(y)  and f(x+y)=f(x)f(y) doesn′t mean   f′(x)=f(x).

$${generally} \\ $$$${for}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)\:{we}\:{have}\: \\ $$$${f}\left({x}\right)={a}^{{x}} \:\left({a}>\mathrm{0}\right) \\ $$$${f}'\left({x}\right)=\left(\mathrm{ln}\:{a}\right){a}^{{x}} \neq{f}\left({x}\right)\:\left({except}\:{a}={e}\right) \\ $$$${therefore} \\ $$$${f}'\left({x}\right)={f}\left({x}\right)\:{doesn}'{t}\:{mean}\: \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$$${and}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)\:{doesn}'{t}\:{mean}\: \\ $$$${f}'\left({x}\right)={f}\left({x}\right).\: \\ $$

Commented by mr W last updated on 07/Nov/21

you can′t prove because it′s not true.  example f(x)=10e^x .  you have f′(x)=10e^x =f(x)  but f(x+y)=10e^(x+y)   f(x)f(y)=10e^x ×10e^y =100e^(x+y)   f(x+y)≠f(x)f(y)

$${you}\:{can}'{t}\:{prove}\:{because}\:{it}'{s}\:{not}\:{true}. \\ $$$${example}\:{f}\left({x}\right)=\mathrm{10}{e}^{{x}} . \\ $$$${you}\:{have}\:{f}'\left({x}\right)=\mathrm{10}{e}^{{x}} ={f}\left({x}\right) \\ $$$${but}\:{f}\left({x}+{y}\right)=\mathrm{10}{e}^{{x}+{y}} \\ $$$${f}\left({x}\right){f}\left({y}\right)=\mathrm{10}{e}^{{x}} ×\mathrm{10}{e}^{{y}} =\mathrm{100}{e}^{{x}+{y}} \\ $$$${f}\left({x}+{y}\right)\neq{f}\left({x}\right){f}\left({y}\right) \\ $$

Commented by mahdipoor last updated on 07/Nov/21

thanks

$${thanks} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com