Calc Question

We have a function f that's defined by $f(x)=e^x \cos x$, where the domain is $[0, 2\pi]$ and we need to find the x-coordinate of each point of inflection.

I found the first derivative to be $f’(x)=e^x(\cos x- \sin x)$ by using the product rule and then pulling out the e^x from both factors.

I found the second derivative to be $f’’(x)=-2e^x \sin x$ by using the product rule again from the result of the first derivative, again pulling out the e^x. The +cosx cancelled with the -cosx and the two -sinx combined.

I set the 2nd derivative equal to 0 to get $x = 0, \pi$ which in theory should be my points of inflection. However, 0 is an endpoint. 

Can an endpoint be a point of inflection??

Also, if anyone reading this could check my work, I would be very appreciative. Thanks.