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August 21st, 2014, 01:15 PM  #1 
Newbie Joined: Aug 2014 From: England Posts: 2 Thanks: 1  Determining the nature of a stationary point?
Hello! I think I must be losing my mind because I cannot figure out a perfectly simple problem. I am supposed to find the coordinates of, and determine the nature of, the stationary point(s) of the curve: y=(2x3)^6 So differentiating it you get dy/dx = 12(2x3)^5 So there is a single stationary point at (1.5,0) and differentiating it again you get d^2y/dx^2 = 120(2x3)^4 So at x=1.5 you get that the second derivative is equal to zero, so that means it's a point of inflexion, right? But it is actually a minimum point... Would be very grateful if someone could point out where I am going wrong! Thanks in advance 
August 21st, 2014, 01:23 PM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 
What about $\displaystyle y = x^4$?

August 21st, 2014, 01:25 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs 
The second derivative being equal to zero is only one requirement for an inflection point. The other is that the sign of the second derivative must change across that point. Is this the case here?

August 22nd, 2014, 12:41 AM  #4 
Newbie Joined: Aug 2014 From: England Posts: 2 Thanks: 1 
I see... So I guess whenever you get a second derivative of zero you have to check what it means by seeing what is going on on either side... Thank you 

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determining, nature, point, stationary 
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