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 $\arccos(\frac{1}{2})$?