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.
Your works looks right. As for your question, you should not include endpoints as points of inflection.
