Curve a - $f'''$ Curve b - $f''$ Curve c - $f'$ Curve d - $f$
Work Step by Step
First look at curve d. Curve d has 2 local extrema and 3 times passes the $Ox$ line. If curve d represents $f'$, $f''$ or $f'''$, there must exist one curve which represents $f$ that has 3 local extrema. However, none of the remaining curves has up to 3 local extrema. Therefore, curve d must represents $f$. Since curve d represents $f$ and has 2 local extrema, there must exist another curve which represents $f'$ that passes the $Ox$ line 2 times. In the remaining curves, only curve c passes the $Ox$ line 2 times. So curve c represents $f'$. Now look at curve c. Curve c has 1 local extrema. At first, it goes down very steepily, then less and less steepily before almost lies horizontally. For the next part, it goes up at first slightly, then increasingly steepily. From the description above, we can deduce that the graph of $f''$ would at first be negative, then it passes the $Ox$ line to become more and more positive till the end. Only curve b fits this description. So curve b represents $f''$. That leaves curve a representing $f'''$.