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I have this equation:

$9x + \cos x = 0$ but I need to write out and prove why it has one real root. Could someone maybe give me a few pointers or what do I do exactly?

1 Answers 1

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Let $f(x)=9x+\cos x$ then $f$ is differentiable and $f'(x)=9-\sin(x)>0$.
So $f$ is strictly increasing, moreover $\displaystyle\lim_{-\infty}f=-\infty$ and $\displaystyle\lim_{+\infty}f=+\infty$ so $f(x)=0$ has exactly one solution (there is a solution by the IVT and the solution is unique by the monotony).

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    Thank $y$ou for your commen$t$. I corrected my answer. PS : I didn't know the word "nitpicking", thank you.2012-07-02