Question

For each sign chart for $f^{\text{'}}$in Exercises 51–56, sketch possible graphs of both $f^{\text{'}}$and $f$. On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$is zero and $x$-values where $f^{\text{'}}\left(x\right)$does not exist are indicated by tick-marks labeled “DNE.”

Short answer

The possible picture of $f^{\text{'}}$is,

The possible picture of $f$is,

Step-by-step solution

Step 1. Given information

On the sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$is zero and $x$-values where $f^{\text{'}}\left(x\right)$does not exist are indicated by tick-marks labeled “DNE.”

Step 2. From the sign chart, it is cleared that the value of f' is positive at -∞,-2 and negative at -2,∞.

Therefore, the possible picture of $f^{\text{'}}$is shown below:

Step 3. As f' is positive on -∞,-2, f is increasing on -∞,-2.

As $f^{\text{'}}$is negative on $\left(-2,\infty \right)$, $f$is decreasing on $\left(-2,\infty \right)$.

Therefore, the possible picture of $f$is,