Q. 52 - Calculus

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.”

Step-by-Step Solution

Answer

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

The possible picture of $f$ is,

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,