Seeing as the first derivative of $f$ is a function in its own right, $\fd{f}$ must have its own first derivative. The first derivative of $\fd{f}$ is the second derivative of $f$ and is symbolized as $\sd{f}$ ($f$ double-prime). Likewise, $\td{f}$ ($f$ triple-prime) is the first derivative of $\sd{f}\text{,}$ the second derivative of $\fd{f}\text{,}$ and the third derivative of $f\text{.}$

All of the graphical relationships you've established between $f$ and $\fd{f}$ work their way down the derivative chain; this is illustrated in Table 4.4.1.

 When $\fd{f}$ is … $f$ is … When $\sd{f}$ is … $\fd{f}$ is … When $\td{f}$ is … $\sd{f}$ is … Positive Increasing Negative Decreasing Constantly Zero Constant Increasing Concave Up Decreasing Concave Down Constant Linear

# Subsection4.4.1Exercises

##### 1

Extrapolating from Table 4.4.1, what must be true about $f$ over intervals where $\sd{f}$ is, respectively: positive, negative, or constantly zero?

##### 2

A function, $g\text{,}$ and its first three derivatives are shown in Figures 4.4.24.4.5, although not in that order. Determine which curve is which function ($g,\fd{g},\sd{g},\td{g}$).

Three containers are shown in Figures 4.4.64.4.8. Each of the following questions are in reference to these containers.

##### 3

Suppose that water is being poured into each of the containers at a constant rate. Let $h_a\text{,}$ $h_b\text{,}$ and $h_c$ be the heights (measured in cm) of the liquid in containers 4.4.64.4.8 respectively, $t$ seconds after the water began to fill the containers. What would you expect the sign to be on the second derivative functions $\sd{h_a}\text{,}$ $\sd{h_b}\text{,}$ and $\sd{h_c}$ while the containers are being filled? (Hint: Think about the shape of the curves $y=\fe{h_a}{t}\text{,}$ $y=\fe{h_b}{t}\text{,}$ and $y=\fe{h_c}{t}\text{.}$)

##### 4

Suppose that water is being drained from each of the containers at a constant rate. Let $h_a\text{,}$ $h_b\text{,}$ and $h_c$ be the heights (measured in cm) of the liquid remaining in the containers $t$ seconds after the water began to drain. What would you expect the sign to be on the second derivative functions $\sd{h_a}\text{,}$ $\sd{h_b}\text{,}$ and $\sd{h_c}$ while the containers are being drained?

##### 5

During the recession of 2008–2009, the total number of employed Americans decreased every month. One month a talking head on the television made the observation that “at least the second derivative was positive this month.” Why was it a good thing that the second derivative was positive?

##### 6

During the early 1980s the problem was inflation. Every month the average price for a gallon of milk was higher than the month before. Was it a good thing when the second derivative of this function was positive? Explain.