[tex]\begin{gathered} Inflection\:points: \\ (\frac{\pi}{2},\frac{\pi}{2}) \\ (\frac{3\pi}{2},\frac{3\pi}{2}) \\ Concave\:downward:\lbrack0,\frac{\pi}{2}\rbrack\cup(\frac{3\pi}{2},2\pi\rbrack \\ Concave\:upward:(\frac{\pi}{2},\frac{3\pi}{2}) \end{gathered}[/tex]
1) Since we need to find the inflection points, we need to take the second derivative of this function, and check whether f(x) is equal to zero or undefined.
[tex]\begin{gathered} f"(x)=\frac{d^2}{dx^2}\left(x+5\cos\left(x\right)\right) \\ \frac{d}{dx}(x+5cos(x)=1-5sin(x) \\ \frac{d}{dx}\left(1-5\sin \left(x\right)\right)=-5\cos \left(x\right) \\ f^{\prime}^{\prime}(x)=-5cos(x) \\ -5\cos \left(x\right)=0,\:0\le \:x\le \:2\pi \\ \frac{-5\cos \left(x\right)}{-5}=\frac{0}{-5} \\ \cos \left(x\right)=0 \\ \end{gathered}[/tex]
2) Now, we need to find the solutions for cos(x) within the given interval:
[tex]\begin{gathered} cos(x)=0,0\leq x\leq2\pi \\ x=\frac{\pi}{2},\:x=\frac{3\pi}{2} \end{gathered}[/tex]
3) The next step is to find the y-coordinate, so let's plug each value of x we have just found into the original function:
[tex]\begin{gathered} f(x)=x+5cos(x) \\ f(\frac{\pi}{2})=(\frac{\pi}{2})+5cos(\frac{\pi}{2})\Rightarrow f(\pi/2)=\frac{\pi}{2} \\ \\ f(\frac{3\pi}{2})=\frac{3\pi}{2}+5cos(\frac{3\pi}{2})=\frac{3\pi}{2} \end{gathered}[/tex]
So the point of inflections are:
[tex]\begin{gathered} \left(\frac{\pi}{2},\frac{\pi}{2}\right) \\ \left(\frac{3\pi}{2},\frac{3\pi}{2}\right) \end{gathered}[/tex]
4) The Concavity can be found by combining the Domain with the inflection points, or we can also check them geometrically:
So, we can tell that:
[tex]\begin{gathered} Concave\:downward:\lbrack0,\frac{\pi}{2}\rbrack\cup(\frac{3\pi}{2},2\pi\rbrack \\ Concave\:upward:\:(\frac{\pi}{2},\frac{3\pi}{2}) \end{gathered}[/tex]
