Show there's only one root in an interval 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 $[2,2.663]$?

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?
