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August 21st, 2014, 12:15 PM   #1
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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=(2x-3)^6

So differentiating it you get

dy/dx = 12(2x-3)^5

So there is a single stationary point at (1.5,0)

and differentiating it again you get

d^2y/dx^2 = 120(2x-3)^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
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August 21st, 2014, 12:23 PM   #2
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What about $\displaystyle y = x^4$?
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August 21st, 2014, 12:25 PM   #3
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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?
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August 21st, 2014, 11:41 PM   #4
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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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