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Critical points: Write the answers in increasing order, separated by commas.

Evaluate $f$ at each of those critical points. Write your answers as ordered pairs of the form $(a,b)$, where $a$ is the critical point and $b$ is the value of $f$ at the critical point. For first critical point: For second critical point:

Plot critical points on the above graph, i.e., plot the points $(a,b)$ you just calculated.

Online, to plot the critical points, drag the slider $n_{cp}$ to specify the number of critical points and move the large blue points to the proper locations.

Left interval: Middle interval: Right interval:

Online, if you need to enter $\infty$, write it as oo or as the symbol ∞. Write $-\infty$ as -oo or using the symbol as -∞.

oo

∞

-oo

-∞

On left interval $f$ is decreasing increasing On middle interval $f$ is increasing decreasing On right interval $f$ is decreasing increasing

Roots of $f$:

What is the value of $f$ at each of those roots?

Plot the roots of $f$ on the above graph.

(Usually, the next step is to sketch the graph of $f$ from this information, but in this warm-up problem, you have the graph of $f$.)

The critical points are $x=$ and $x=$ . Enter in increasing order.

Calculate the value of $f$ at the critical points. $f=$ and at the first and second critical points, respectively.

For the left interval, $f$ is decreasing increasing For the middle interval, $f$ is decreasing increasing For the right interval, $f$ is increasing decreasing

Roots of $f$: $x=$ Separate answers by commas.

Online, we don't have a way for you to sketch the function. Instead, drag $n_{cp}$ to specify the number of critical points and move the large blue points to the proper locations. Drag $n_z$ to specify the number of zeros (or roots) and move the small red points to the proper locations. If these points are in the correct locations, when you click “sketch function,” the applet will show the graph of the function. If the points are not in the correct locations, the applet will just connect the points with line segments.

$x=$ Enter answers in increasing order, separated by commas.

Find the value of $f$ at the critical points: Enter answers in the same order as above, separated by commas.

Left interval: Right interval:

In the left interval, $f$ is increasing decreasing In the right interval, $f$ is decreasing increasing

Online, we don't have a way to sketch the graph.

$f'(x) =$ if $x >$ $f'(x) =$ if $x <$ $f'(x)$ does not exist if $x=$

Critical points: $x=$ Enter answers in increasing order, separated by commas.

Left interval: $f$ is decreasing increasing on the interval Right interval: $f$ is increasing decreasing on the interval

Critical points: $x=$ Enter in increasing order, separated by commas.

Calculate the values of $f$ at the critical points: Enter in same order as the critical points, separated by commas.

Left interval: $f$ is increasing decreasing on the interval Right interval: $f$ is increasing decreasing on the interval

$f(10) =$ , $f(-6)=$ .