Chapter 2.7, Problem 41E

To determine

To identify: The curves $f,f^{\prime }$ and ${f^{\prime }}^{\prime }$ on given graph with proper explanation.

Answer to Problem 41E

In the given graph,$a=f$, $b=f^{\prime }$ and $c=f^{″}$.

Explanation of Solution

Graph:

The given graph is shown as in Figure 1,

Observation:

From the Figure 1, it is observed that the point where $c\left(x\right)=0$ is the same point where graph of $b\left(x\right)$ has horizontal tangent.

Recall that the derivative of a function is zero where the function has a horizontal tangent.

Therefore, $c\left(x\right)$ is the derivative of the graph $b\left(x\right)$. That is, $b^{\prime }\left(x\right)=c\left(x\right)$.

Observe the graph of a and b carefully.

The point where $b\left(x\right)=0$ is the same point where graph of $a\left(x\right)$ has horizontal tangent.

Recall that the derivative of a function is zero where the function has a horizontal tangent.

Therefore, $b\left(x\right)$ is the derivative of the graph $a\left(x\right)$. That is, $a^{\prime }\left(x\right)=b\left(x\right)$.

From Figure 1, it is observed that, neither $a\left(x\right)$ nor $b\left(x\right)$ are equal to zero where the graph of $c\left(x\right)$ has horizontal tangent. Clearly, $f^{″}=c\left(x\right)$.

Use the above information $a^{\prime }\left(x\right)=b\left(x\right)$ and $b^{\prime }\left(x\right)=c\left(x\right)$ to concluded that $a^{″}\left(x\right)=c\left(x\right)$.

Thus, $a=f$,$b=f^{\prime }$, and $c=f^{″}$.