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

I've always thought that the points where the f(x) graph cuts the x-axis is where the points of inflection for f'(x) are.
