# What does third derivative tell about inflection point?

I was trying to find the nature (maxima,minima,inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$

But I faced a conceptual problem.It is given in the solution to the problem that $f''(0)=0$ and $f'''(0) \neq 0$ so $0$ is not an inflection point.But why should we check the third derivative?

Isn't checking first and second derivative sufficient for verifying an inflection point ? Why must the higher order odd derivatives be zero for an inflection point?

• What is your definition of inflection point? I think that this is important. – Crostul Feb 5 '16 at 12:02
• Where curve changes its concavity... – user220382 Feb 5 '16 at 12:04
• In this case, $\;f''(0)=f'''(0)=0\;$ , but $\;f^{(iv)}(0)=-2\neq0\implies x=0\;$ is not an extremum point of $\;f'(x)\;\implies x=0\;$ is not an inflextion point. – DonAntonio Jan 12 at 21:58

An inflection point is where a curve changes from concave to convex or vice versa. There are two types of inflection points: stationary and non-stationary. Stationary means that at this point the slope (thus $f'$) is $0$. These points are also called saddle-points.

Non-stationary inflection points are different. They are where the slope is at maximum, i.e. you have to maximize $f'$ in order to find them. You know from caculus that you need to look at both the $f'$ and $f''$ derivatives to determine whether a $f$ has a maximum. But now, since we look for maxima of $f'$, we have to look at $f''$ and $f'''$.

• Wait a second.For $f'(x)=0 , f''(x)=0$ and $f'''(x) \neq 0$ tells us that it is a stationary turning point or non stationary? – user220382 Feb 5 '16 at 12:10
• $f'' = 0$ must be too, else it's a maximum/minimum – Adrian Feb 5 '16 at 12:13
• Sorry sorry.Look at my question now.Please. – user220382 Feb 5 '16 at 12:14
• i meant, it's stationary, else it's a maximum/minimum – Adrian Feb 5 '16 at 12:15

It depends on your definition of inflection point. You have given this as

"Where curve changes its concavity".

In this case checking $f'''(x)$ is not necessary since for example,

$$f(x) = x^3 \text{ gives } f'''(x) = 6 \neq 0$$ however $x = 0$ is an inflection point since there is a change in concavity here.