How can you find the $x$-coordinate of the inflection point of the graphs of $f'(x)$ and $ f''(x)$?

Question

So I understand how to find the inflection points for the graph of $f(x)$.

But basically, I have been shown a graph of an example function $f(x)$ and asked the state the inflection points of the graph. (I am just shown a curve... not given an actual function in terms of $x$)

Easy enough... I can see by looking where the concavity of the curve changes. But here is the tricky part. I am then asked to find the inflection points of the curves of $f'(x)$ and $f''(x)$.

From the graph of $f(x)$, I can draw in the values it cuts the $x$-axis, and whether it is positive/negative, but I don't understand how you can also comment on the inflection points from it?

i.e. $f(x)$ has both absoulute and local max values at $x = 2$ and $x = 6$, and local min at $x = 4$. Hence, $f'(x)$ cuts $x$ axis at $2, 4$ and $6$. I would imagine the local min/max would therefore be $3$ and $5$ (middle of $2$ & $4$, and $4$ & $6$ respectively)

The book gives answers a) $(3,5)$ Got it! b) $2, 4, 6$ c)$1, 7$

How do you find the point of inflection for $f'(x)$ and $f"(x)$ in this case?

Answer 1

An inflection point is where the second derivative is zero and changes sign. Thus, if the graph is of the second derivative then notice that the zeroes with sign change are at 1 and 7 on that graph.

If the graph is of the first derivative then the local min/max will be the inflection points since this is where if one were to look at the derivative of this graph it would be the second derivative that goes to zero at the extreme point as well as the sign change that is the criteria.

Answer 2

Consider only ordinary inflexions, i.e. $\mathrm{f}''(x_0)=0$, but $\mathrm{f}'''(x_0)\neq 0$

We can recognise an ordinary inflexion as a point where the curve cross the tangent line.

At almost all points a regular curve has two-point contact with its tangent line, e.g. like the curve $y=x^2$ does with the line $y=0$. The curve stays on the same side of the tangent line at such points.

At ordinary inflexions it has tree-point contact, e.g. like the curve $y=x^3$ does with the line $y=0$. The curve crosses the tangent line at such points.

The curve below has an ordinary inflexion at $B$. That is the only inflexion. All points close to $A$ have the curve below the tangent line. All points close to $C$ have the curve above the tangent line. The point $B$ is the inflexion. The curve crosses its tangent line at $B$.

Things get complicated when there are higher inflexions. The curve $y=x^4$ has a higher inflexion at $x=0$. However, the curve has four-point contact with the tangent line and does not cross the tangent line. In general, a curve only crosses its tangent line when it has odd-point contact.