Proving that f(x)=kecx without Assuming f=0

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SUMMARY

The discussion centers on proving that the function f(x) can be expressed as f(x) = kecx, where k is a constant, under the condition that f' = cf for some constant c, without assuming that f is never zero. The key insight is to demonstrate that f cannot equal zero at the endpoints of an open interval where it is not near zero. The hint suggests utilizing the properties of logarithmic differentiation and potentially the Mean Value Theorem to derive the solution.

PREREQUISITES
  • Understanding of differential equations, specifically the equation f' = cf.
  • Knowledge of logarithmic differentiation and its application.
  • Familiarity with the Mean Value Theorem in calculus.
  • Basic concepts of continuity and limits in real analysis.
NEXT STEPS
  • Study the application of logarithmic differentiation in solving differential equations.
  • Research the Mean Value Theorem and its implications for function behavior on intervals.
  • Explore the properties of exponential functions and their derivatives.
  • Investigate the conditions under which a function can be zero and the implications for continuity.
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators seeking to enhance their understanding of function behavior and proofs in real analysis.

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Homework Statement



Suppose that on some interval the function f satisfies f'=cf for some number c. Show that f(x)=kecx for some k without the assumption that f is never 0. Hint: Show that f can't be 0 at the endpoint of an open interval on which it is nowhere hear 0

Homework Equations



f'=cf
(log (abs f))'=f'/f

The Attempt at a Solution



i tried to use the case where f=0, which meant that f'=0. From there, it follows that f(x)=k. That's where i got stuck. I know somehow I have to show that f can be f(x)=kecx=0, where k=0, but how? Perhaps I need to figure out how to use the hint. Does the hint have anything to do with Mean Value Theorem?
 
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Integrate f'/f and then solve for f.
 

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