Second derivatives and inflection points

[keroberous]

Hi there. I'm having some trouble wrapping my head around some ideas of inflection points as they relate to the second derivative.

I know that an inflection point occurs when f''(x)=0 in most cases. This makes sense to me because at this inflection point the slopes of the tangent change from increasing to decreasing, so the rate of change of the changing slopes must be zero. This is the same reasoning why a turning point occurs when f'(x)=0 (the original function f changes from increasing to decreasing, or vice versa).

However, in some cases (like f(x)=x^4) it's possible for both the first and second derivatives to be zero at the same place and I'm not sure as to what the reasoning behind this is. In this case, f'(0)=0 meaning there is a turning point here, but f''(0)=0 also but this is not an inflection point. It makes sense that there shouldn't be an inflection point at a turning point, but then why is the second derivative zero? In this case as x increases, the slope of the tangent is always increasing, so by that logic shouldn't the second derivative never equal zero?

Thanks for any help here!

 

[fresh_42]

In order to ...

at this inflection point the slopes of the tangent change from increasing to decreasing

... you also need the next derivative to make sure there is a change. Otherwise, it could go from increasing to zero and back to increase. The next derivative (unequal zero) fixes the change. This, however, is not the case for $x\longmapsto x^4.$

The second derivative is the curvature. You need the third (unequal zero) to have a change in curvature. Otherwise, it goes from e.g. left-handed to zero and back to left-handed.

 

[keroberous]

Thanks for your answer. I'm still not clear as to why exactly the second derivative at zero for x^4 should be zero. The function is clearly concave up at this point (isn't it) so should it not be a positive second derivative?

Also, I'm not quite sure what the difference is in the changing tangents at the minimum value of x^4 compared to the local minimum of a cubic polynomial, for example. In both cases, the slope of the tangent is increasing (from negative slopes to positive slopes) as you move through this minimum point, but why is the second derivative of this point in the cubic positive where the second derivative at this point in x^4 is zero?

 

[fresh_42]

The functions $x\longmapsto x^n$ are especially flat at $x=0.$ Look at the plots on e.g. WolframAlpha:
https://www.wolframalpha.com/input/?i=y=x^21

\begin{align*}
x\longmapsto x^2\, &: \,\text{horizontal tangent, curvature positive}\\
x\longmapsto x^3\, &: \,\text{horizontal tangent, zero curvature, curvature change}\\
x\longmapsto x^4\, &: \,\text{horizontal tangent, zero curvature, no curvature change, jerk positive}\\
x\longmapsto x^5\, &: \,\text{horizontal tangent, zero curvature, no curvature change, no jerk change}\\
\ldots &\ldots
\end{align*}

Here is a nice picture in case $f(x)$ describes a physical motion of a body:
https://en.wikipedia.org/wiki/Jerk_(physics)

 

[Office_Shredder]

I think focusing on the zeros for intuition is where you're getting tripped up.

An inflection point is where the sign of f''(x) changes. If f'' is continuous, it must change where f''(x)=0. But there are plenty of examples where f''(x)=0 and the sign of f'' doesn't change when you cross that value of x. So just knowing f''(x)=0 isn't enough to tell you it's an inflection point, it's just a good test to find the places that might be inflection points.

 

[snorkack]

In order to ...

... you also need the next derivative to make sure there is a change. Otherwise, it could go from increasing to zero and back to increase. The next derivative (unequal zero) fixes the change. This, however, is not the case for $x\longmapsto x^4.$

The second derivative is the curvature. You need the third (unequal zero) to have a change in curvature. Otherwise, it goes from e.g. left-handed to zero and back to left-handed.

No. You need the first nonzero derivative after first to be odd. Might be third derivative, might be fifth derivative or fifteenth. All of these are inflection points.
If first nonzero derivative after first is even, whether fourth, sixth et cetera, the curvature has the same sign both sides.