Hello. I have the following query. Consider a continuous function f(x). We want to know whether this function ever gets zero or not. So we make the assumption that it does. If (e.g) we have [itex]f(x)=x^2 + 4x + 2 - cosx[/itex](adsbygoogle = window.adsbygoogle || []).push({});

we'll assume that [itex]f(x)=0 (=) x^2 + 4x + 2 -cosx =0 [/itex]

now we differentiate with respect to x

[itex]2x + 4 +sinx =0[/itex] and again

[itex]2 + cosx = 0 (=) cosx = -2 [/itex] which is impossible.

Right now we have proven that f ''(x) can't be zero. Is there any theorem that, given some prerequisites for f(x), can show that f(x) can't be zero too, using the above demonstrated way?

Thanks in advance, I hope you understand the core of my question. (I am not looking for the Bolzano Theorem, btw)

Edit1: Another way of asking this is : Is there any theorem that proves, given prerequisites, that if [itex]f^n (x)=0 then f(x)=0[/itex], I won't specify how many (n) times differentiated.

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# Proving that a certain f(x) can't be zero by differentiating

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