Recently I have been struggling to solve the following problem:
The figure shows the graphs of $f$, $f'$, and $f''$. Identify each curve, and explain your choices.
I know the answer, but I have no idea how to explain this. Any help would be appreciated...
To solve this problem, some geometrical and intuitive description of the derivative of a function would be useful since we have no algebraic description of the depicted functions.
Given a function $f(x)$, the derivative of $f$ with respect to $x$, denoted here $f^{\prime}$, describes at a given point $x_0$ the slope of a line that is tangent to $f$ at $x_0$. This can be rephrased as follows. Consider the tanget line to $f$ at $x_0$. Since it is a linear function we can describe it as $y = kx + q$ for some $k,q \in \mathbb{R}$. Observe that the tanget line changes as we move along the graph of $f$ which implies that the coefficients $k$ and $q$ (where $k$ is the relevant one for us) are functions of variable $x$. So, to a given differentiable function $f$ we can associate a family of tangent lines $y_x = k(x)x + q(x)$ and the $k(x)$ is precisely $f^{\prime}(x)$. Then $f^{\prime}$ evaluated at $x_0$ is precisely the value of $k$ in the description of the tangent line, i.e. $f^{\prime}(x_0) = k$.
The first derivative thus contains the information of whether $f$ is constant ($f^{\prime} = 0$), increases ($f^{\prime} > 0$) or decreases ($f^{\prime} < 0$): the derivative of a function evaluated at a point in which the function is increasing cannot have negative value, that is to say $$f \text{ increasing at } x_0 \Rightarrow f^{\prime}(x_0) > 0 \ .$$ This is precisely what Michael Lee meant when explaining in his comment that the graph $a$ cannot correspond to the derivative of either $b$ or $c$ because there is a region in which both $b$ and $c$ are increasing yet $a$ is negative. Hence $a$ must be the graph of $f$.
We can apply the same thinking to the second derivative and understand $f^{\prime \prime}$ to be the function that describes the increasing and decreasing of $f^{\prime}$ though this is unnecessary for the problem at hand. Since (as was also mentioned in the comments) there is a region around the origin in which $a$ is increasing (thus $f^{\prime}$ cannot have negative values) and $c$ is negative. We conclude that $b$ must correspond to the graph of $f^{\prime}$ and $c$ to the graph of $f^{\prime \prime}$.