Question Number 211486 by mnjuly1970 last updated on 10/Sep/24
![Mathematical Analysis... f:R → R is diffrentiable function, f and f ′ , has no common zero on R . prove that the following set is finite. X = { x ∈ [0 , 1 ] ∣ f(x)=0 } −−−−−−−−−−−−](https://www.tinkutara.com/question/Q211486.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{Mathematical}\:\:\:\:\:{Analysis}… \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:{f}:\mathbb{R}\:\rightarrow\:\mathbb{R}\:{is}\:{diffrentiable}\:{function}, \\ $$$$\:\:\:\:\:{f}\:\:{and}\:\:{f}\:'\:,\:{has}\:{no}\:{common}\:{zero} \\ $$$$\:\:\:\:\:{on}\:\:\mathbb{R}\:. \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:{the}\:{following}\:{set}\:{is}\:{finite}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{X}\:=\:\left\{\:{x}\:\in\:\left[\mathrm{0}\:,\:\mathrm{1}\:\right]\:\mid\:{f}\left({x}\right)=\mathrm{0}\:\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−−−−− \\ $$