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I want to prove that $-3x+7 = \cos x$ only has one root on $\mathbb R$. I've already shown that there is one root in the interval $[2,3]$ using the Intermediate Value Theorem, and I can show that there are no roots wherever $-3x+7 \notin [-1,1] \because \cos x \in [-1,1] \ \forall \ x \in \mathbb R$. How can I show there is only one root in the interval $[2,2.663]$?

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If you look at the function $f(x) = -3x+7-\cos x $ have the derivative $f'(x) = -3 +\sin x $ what can you say about the sign of the derivative? what does that means?

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    $\begingroup$ Ah, so since the derivative is always negative then $f(x)$ is always decreasing, and when it intersects the $x$ axis it will never return. $\endgroup$
    – jeremy909
    Commented Sep 8, 2020 at 19:04

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