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. The graph of f' should be positive on (-∞,1),(3,8),(8,∞) and negative on 1,3.

It should also have roots at $x=1$and $x=8$and $f^{\text{'}}$should be undefined at $x=3$.

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

Step 3. The graph of f should be increasing on (-∞,1),(3,8),(8,∞).

It decreasing on $\left(1,3\right)$. It should also have horizontal tangent lines at $x=1$(a local maximum) and $x=8$(not an extremum) and $f$should be nondifferentiable at$x=3$ (a local minimum).

The possible picture of $f$is given by,