use the graph to answer the following questions. (a) over which intervals is the function decreasing? choose…

Explanation:

Step1: Identify decreasing intervals

A function is decreasing when its graph goes down - ward as we move from left - to - right. From the graph, we can see the function is decreasing on $(-\infty,-1)$, $(1,4)$ and $(6,\infty)$.

Step2: Locate local minima

Local minima occur at points where the function changes from decreasing to increasing. From the graph, local minima are at $x = - 1$ and $x = 4$.

Step3: Determine sign of leading coefficient

Since the function goes to $-\infty$ as $x\to\pm\infty$, the leading coefficient is negative. For an even - degree polynomial, the function goes to $\infty$ as $x\to\pm\infty$ if the leading coefficient is positive and $-\infty$ as $x\to\pm\infty$ if the leading coefficient is negative. For an odd - degree polynomial, the function goes to $\infty$ as $x\to\infty$ and $-\infty$ as $x\to-\infty$ (or vice - versa) depending on the sign of the leading coefficient. Here, since it goes to $-\infty$ in both directions, the degree must be even and the leading coefficient negative.

Step4: Estimate degree of the function

The number of turning points of a polynomial function of degree $n$ is at most $n - 1$. The graph has 4 turning points. So the degree $n$ satisfies $n-1\geq4$, or $n\geq5$. Also, since the function goes to $-\infty$ as $x\to\pm\infty$, the degree $n$ is even. So possible degrees are odd numbers greater than or equal to 5.

Answer:

(a) $(-\infty,-1),(1,4),(6,\infty)$ (b) $-1,4$ (c) Negative (d) 5,7,9