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# Calc Question

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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