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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.

 Nov 30, 2022
 #1
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Your works looks right.  As for your question, you should not include endpoints as points of inflection.

 Nov 30, 2022
 #2
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I agree with guest. Your works looks about right, I would check 0 if it is an answer, im not that advanced, i honestly dont know if and endpoint could be a point of inflection, sorry!

 Nov 30, 2022
 #3
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And NO an endpoint CANNOT be a point of inflection

 Nov 30, 2022
edited by Imcool  Nov 30, 2022

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