Assume that $y=f_1(t)$ and $y=f_2(t)$ are two solutions of the following function: $$\frac{\mathrm d y}{\mathrm d t}=\mathrm e^{t^3}- \mathrm e^{t^4}$$ and $f_1(0)>f_2(0)$. How can I describe the signs in $f_2(1)-f_1(1)$ using the fundamental theorem of calculus? I have no clue where to start.

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    $\begingroup$ If two functions have the same derivative, the difference of the two functions is… $\endgroup$ – egreg Nov 27 '14 at 12:42
  • $\begingroup$ @egreg: You should make this an answer, despite its shortness. I cannot improve it! $\endgroup$ – Rory Daulton Nov 27 '14 at 15:01
  • $\begingroup$ @studyhenry Did you get the solution for the sign? $\endgroup$ – mvw Nov 27 '14 at 15:12

By definition, $f_1'(t)=f_2'(t)$. So the difference $$ g(t)=f_1(t)-f_2(t) $$ has zero derivative and so it is …

  • $\begingroup$ thanks for your answer. However, IMO, $f_1$ and $f_2$ aren't same since $f_1(0) > f_2(0)$. I think the answer lies somewhere by using the FTC. Still researching myself... $\endgroup$ – studyhenry Nov 28 '14 at 7:45
  • $\begingroup$ @studyhenry Yes, they aren't the same, but the difference is constant. $\endgroup$ – egreg Nov 28 '14 at 9:25

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