Q. 56 - 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(x) 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 . The graph of f ' should be positive on - 1 , 4 , 4 , ∞ and negative on - ∞ , - 1 .

Also, at $x=-1,4$ is undefined.

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

Step 3 . As f ' is positive on - 1 , 4 , 4 , ∞ , f is increasing on - 1 , 4 , 4 , ∞ .

As $f^{\text{'}}$ is negative on $\left(-\infty ,-1\right)$,$f$ is decreasing on $\left(-\infty ,-1\right)$.

The possible picture of $f$ is,