Finding maxima and minima of a function A couple problems are giving me trouble in finding the relative maxima/minima of the function.  I think the problem stems from me possibly not finding all of the critical numbers of the function, but I don't see what I missed.
Given $f(x)= 5x + 10 \sin x$, I calculated the derivative as $5 + 10 \cos x$, and found the first critical number by this work:
$$5+ 10 \cos x=0$$
$$\frac{5}{5}+10 \cos x= 0-5 \Rightarrow 10 \cos x= -5$$
$$\frac{10 \cos x}{10}= \frac{-5}{10}\Rightarrow \cos x= -\frac{1}{2}$$
$$x= \arccos(-\frac{1}{2}) = \text{First critical number is }\frac{2\pi}{3}$$
That gave me the maxima of the formula, since $$f(\frac{2\pi}{3})= 5(\frac{2\pi}{3})+10 \sin(\frac{2\pi}{3})= \frac{10\pi}{3}+5\sqrt3$$
However, I need the other critical number to calculate the minima.  Should I look for the value of $\arccos(\frac{1}{2})$?

$\cos(x)=-1/2$ if and only if there exists an integer $n$ such that $x=2n\pi+2\pi/3$ or $x=2n\pi+4\pi/3$. Hence any of these real numbers $x$ may be (and in fact, is) a relative maximum or a relative minimum of $f$.
