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.
$(-\infty, -5)$ $(-3, 0)$ $(-5, 0)$ $(3, 6)$ $(6, 8)$ $(8, \infty)$
(b) at which $x$-values does the function have local maxima? if there is more than one value, separate them with commas.

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{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Graph Interpretation" ], "new_concepts": [ "Local Extrema", "Increasing and Decreasing Intervals" ], "current_concepts": [ "Graph Interpretation", "Local Extrema", "Increasing and Decreasing Intervals" ] } ## Identify decreasing intervals from local extrema $$ \begin{aligned} &\text{Local extrema (turning points) occur at } x = -5, -3, 0, 3, 6, 8.\ &\text{The function is decreasing on intervals where the graph goes down from left to right:}\ &(-\infty, -5), \quad (-3, 0), \quad (3, 6), \quad (8, \infty) \end{aligned} $$ ## Identify x-values of local maxima $$ \begin{aligned} &\text{Local maxima occur at the "peaks" of the graph.}\ &\text{Peaks are located at the points: } (-3, 3), \quad (3, 1.5), \quad (8, 1)\ &\text{The corresponding } x\text{-values are: } -3, 3, 8 \end{aligned} $$ ## Question (a) Over which intervals is the function decreasing? Choose all that apply. - [x] $(-\infty, -5)$ - [x] $(-3, 0)$ - [ ] $(-5, 0)$ - [x] $(3, 6)$ - [ ] $(6, 8)$ - [x] $(8, \infty)$ ## Question (b) At which $x$-values does the function have local maxima? If there is more than one value, separate them with commas. -3, 3, 8 { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Local Extrema" ] }

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Identify decreasing intervals from local extrema

Identify x-values of local maxima

Answer:

Question (a)

Question (b)