Q. 55 - 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 . 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,