Question

b) identify the graph that displays f in blue and f in red.
a.
b.
c.
d.
c) using the graphs of f and f, indicate where f is increasing and decreasing. give your answer in the
note: when using interval notation in webwork, remember that:
you use inf for ∞ and -inf for −∞,
and use u for the union symbol.
enter dne if an answer does not exist.

Explanation:

Step1: Recall derivative - function relationship

The function $f(x)$ is increasing when $f^{\prime}(x)>0$ and decreasing when $f^{\prime}(x)<0$. Also, at the local - extrema of $f(x)$, $f^{\prime}(x) = 0$.

Step2: Analyze option A

In option A, when the blue curve ($f$) has a local minimum or maximum, the red curve ($f^{\prime}$) does not cross the $x$ - axis, which is incorrect since $f^{\prime}(x)=0$ at local extrema of $f(x)$.

Step3: Analyze option B

In option B, when the blue curve ($f$) is increasing, the red curve ($f^{\prime}$) is below the $x$ - axis, which is wrong because $f^{\prime}(x)>0$ when $f(x)$ is increasing.

Step4: Analyze option C

In option C, when the blue curve ($f$) has local extrema, the red curve ($f^{\prime}$) crosses the $x$ - axis. Also, when the blue curve ($f$) is increasing, the red curve ($f^{\prime}$) is above the $x$ - axis and when the blue curve ($f$) is decreasing, the red curve ($f^{\prime}$) is below the $x$ - axis. This is consistent with the relationship between a function and its derivative.

Step5: Analyze option D

In option D, when the blue curve ($f$) is increasing, the red curve ($f^{\prime}$) is below the $x$ - axis, which is incorrect as $f^{\prime}(x)$ should be positive when $f(x)$ is increasing.

For part c:

Step1: Identify increasing intervals

For the correct graph (option C), the function $f$ (blue curve) is increasing when $f^{\prime}(x)>0$ (red curve is above the $x$ - axis). From the graph, $f$ is increasing on the intervals $(-\infty, - 2)\cup(2,\infty)$.

Step2: Identify decreasing intervals

The function $f$ is decreasing when $f^{\prime}(x)<0$ (red curve is below the $x$ - axis). From the graph, $f$ is decreasing on the interval $(-2,2)$.

Answer:

b) C
c) Increasing: (-INF, - 2)U(2, INF); Decreasing: (-2,2)