Q. 53 - 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 3 , ∞ and negative at - ∞ , - 3 and - 3 , 0 .

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

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

As $f^{\text{'}}$ is negative on $\left(-\infty ,-3\right)$ and $\left(-3,0\right)$,$f$ is decreasing on $\left(-\infty ,-3\right)$ and $\left(-3,0\right)$.

The possible picture of $f$ is,