Points of Inflection, The Second Derivative Test.

[MisterMan]

Hi, I am having a little trouble understanding something my lecturer said about using the table of signs to check whether there exists a point of inflection when y'' = 0. I understand that in order for there to be a point of inflection at x₀ say, I require to check the value of y'' at either side of the value x₀, and if the value of y'' changes sign ( from positive to negative or vice versa ) then an inflection point exists at x₀.

The problem I am having is that my lecturer said when checking the sign at either side of x₀, you choose values a little bit smaller and a little bit larger than x₀. I don't understand why this is needed, I believe his explanation to be that there might be local maxima/minima that alter the value of y'' or something, but I'm not totally sure if that is what he said or why it is necessary to choose values close to x₀ as even if that was what he said, we calculate the minima/maxima prior to working out the possible inflection points. I hope I have made myself clear and any help is appreciated.

 

[tiny-tim]

Hi MisterMan!

I understand that in order for there to be a point of inflection at x₀ say, I require to check the value of y'' at either side of the value x₀, and if the value of y'' changes sign ( from positive to negative or vice versa ) then an inflection point exists at x₀.

The problem I am having is that my lecturer said when checking the sign at either side of x₀, you choose values a little bit smaller and a little bit larger than x₀. I don't understand why this is needed …

It depends whether you know where all the stationary points (y'' = 0) are …

if you do, then you can check anywhere between that stationary point and the two on either side (or anywhere on one side, if there's no stationary point on that side), since it (obviously) can't change sign between stationary points! …

but if you don't know, you need to be careful, and choose points very close.