Question

below is the graph of a polynomial function with real coefficients. all local extrema of the function are shown in the graph.
use the graph to answer the following questions.
(a) over which intervals is the function decreasing? choose all that apply.
$(-\infty,-9)$ $(-9,-7)$ $(-3,1)$ $(-7,1)$ $(4,7)$ $(7,\infty)$
(b) at which x-values does the function have local minima? if there is more than one value, separate them with commas.
(c) what is the sign of the functions leading coefficient?
(d) which of the following is a possibility for the degree of the function? choose all that apply.
4 5 6 7 8 9

Explanation:

Step1: Determine the decreasing intervals

A function is decreasing when the slope of the tangent line is negative. Looking at the graph, we can see that the function is decreasing in the intervals $(-9,-7)$ and $(4,7)$.

Step2: Identify the local minima

A local minimum is a point where the function changes from decreasing to increasing. From the graph, the \(x -\)values of local minima are \(x=-7\) and \(x = 1\).

Step3: Find the sign of the leading coefficient

For a polynomial function \(y = a_nx^n+\cdots+a_0\), if \(n\) is odd: when \(x\to+\infty\), \(y\to+\infty\) if \(a_n>0\) and \(y\to-\infty\) if \(a_n < 0\); when \(x\to-\infty\), \(y\to-\infty\) if \(a_n>0\) and \(y\to+\infty\) if \(a_n<0\). Since as \(x\to+\infty\), \(y\to+\infty\), the leading coefficient is positive.

Step4: Determine the degree of the polynomial

The maximum number of local extrema of a polynomial function of degree \(n\) is \(n - 1\). The graph has \(4\) local extrema. So \(n-1\geq4\), \(n\geq5\). Also, since the end - behavior (as \(x\to\pm\infty\)) is in opposite directions (because the leading coefficient is positive and degree is odd), the degree \(n\) is odd. So possible degrees are \(5,7,9\).

Answer:

• (a) $(-9,-7),(4,7)$
• (b) $-7,1$
• (c) Positive
• (d) $5,7,9$