Is finding the extreme points of a differentiable function by first derivative always correct?

Question

I came across this question where I was asked to find the local minimum and local maximum of the function $$y=\sec x + 2\ln(|\cos x|),$$ domain of $x$ being $(0,2\pi)-\{\pi/2 , 3\pi/2\}$. I found its first derivative to be $$y=\tan x(\sec x-2)$$ and found that it is zero on three points $\pi/3$, $\pi$ and $5\pi/3$. Then I used the second derivative to see which one is local maximum and which one is local minimum. I found that $\pi$ was the point of local maximum and the other two were points of local minimum. But when I actually saw the graph of the function there was no point of local minimum. There were points on which the value of the function was lesser than that at $\pi/3$ and $5\pi/3$ in fact at $\pi/3$ and $5\pi/3$ the tangent was not even parallel to the $x$ axis. I am not getting where I went wrong. Please help .