[Math] From the graph of the derivative $f'(x)$, make a sketch of the original function $f(x)$ and of the second derivative $f”(x)$

calculusderivativessolution-verification

I haven't started finding the derivatives of functions yet, so at the moment this is strictly about finding the right derivative graph to an original graph. The task was this:

Look at the graph below of the derivative $f'(x)$. From this, make a sketch of the original function $f(x)$ and of the second derivative $f''(x)$.

Task image

In a mail, I was told this as well:

The important part about this question is to match up the features of $f(x)$, $f'(x)$, and $f''(x)$, such as $x$-intercepts, CP's, POI's. There are relationships between these features when comparing a graph with its derivatives, this is what you must demonstrate in your solution.

I'm not quite sure what he means by CP's and I'm guessing POI's means points of interest. Anyway, I ended up with this drawing:

3 graphs

A closer look at the relevant two:

Original f(x) graph
Second derivate

Sorry that they're a little big. My scanner scans with pretty high resolution.

Anyway, does this look right? Did I miss any of the qualities the teacher said it was important to demonstrate?

Best Answer

As a teacher, I would say you did show your understanding of the important features:

  • $f$ is increasing on the intervals where $f'$ is positive.
  • $f$ is decreasing on the intervals where $f'$ is negative.
  • $f$ reaches an extremum at points where $f'$ vanishes.
  • The sign of $f''$ corresponds to the variations of $f'$

It is also very nice that you have drawed $f''$ as linear, meaning you recognized that the graph of $f'$ looks like a parabola.

I would say, very satisfying job.

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