question 22 of 40 (1 point) | question attempt: 1 of unlimited
use the graph to answer the following questions.
your answer
(a) over which intervals is the function decreasing? choose all that apply.
□ $(-\infty, -8)$ □ $(-8, -5)$ □ $(-8, -2)$ □ $(-2, 3)$ □ $(7, 9)$ □ $(9, \infty)$
(b) at which $x$-values does the function have local minima? if there is more than one value, separate them with commas.
try one last time

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Accepted Answer

# Explanation:
## Part (a)
### Step 1: Recall the definition of a decreasing function
A function $ f(x) $ is decreasing on an interval if, for any two points $ x_1 < x_2 $ in the interval, $ f(x_1) > f(x_2) $. Geometrically, this means the graph of the function is falling (going from left to right) over that interval.

### Step 2: Analyze each interval
- **Interval $ (-\infty, -8) $**: The graph is rising (going up from left to right), so the function is increasing here.
- **Interval $ (-8, -5) $**: The graph is falling (going down from $ x = -8 $ to $ x = -5 $), so the function is decreasing here.
- **Interval $ (-8, -2) $**: Wait, no, let's check the graph again. From $ x=-8 $ to $ x=-5 $ it's decreasing, then from $ x=-5 $ to $ x=-2 $ it's increasing (since it goes up to a local maximum at $ x=-2 $). Wait, maybe I misread the intervals. Wait the intervals are $ (-\infty, -8) $, $ (-8, -5) $, $ (-8, -2) $, $ (-2, 3) $, $ (7, 9) $, $ (9, \infty) $. Wait, no, the correct way is to look at the direction of the graph. Let's re - examine:
    - From $ x=-8 $ to $ x=-5 $: The graph is decreasing (moving down).
    - From $ x=-2 $ to $ x = 3 $: The graph is decreasing (moving down from $ x=-2 $ to $ x = 3 $).
    - From $ x = 7 $ to $ x=9 $: The graph is decreasing (moving down from $ x = 7 $ to $ x=9 $). Wait, no, let's check the key points. The local minima and maxima: The local minima are at $ x=-5 $, $ x = 3 $, $ x=9 $? Wait, no, the graph has peaks and valleys. Let's list the critical points (local maxima and minima). The local maxima are at $ x=-2 $, $ x = 7 $ and local minima at $ x=-5 $, $ x = 3 $, $ x=9 $? Wait, no, looking at the graph:
        - Starting from the left, the graph comes from the top, then at $ x=-8 $ it starts to go down (so from $ -\infty $ to $ -8 $ it's increasing), then from $ -8 $ to $ -5 $ it's decreasing (since it goes down to a local minimum at $ x=-5 $), then from $ -5 $ to $ -2 $ it's increasing (goes up to a local maximum at $ x=-2 $), then from $ -2 $ to $ 3 $ it's decreasing (goes down to a local minimum at $ x = 3 $), then from $ 3 $ to $ 7 $ it's increasing (goes up to a local maximum at $ x=7 $), then from $ 7 $ to $ 9 $ it's decreasing (goes down to a local minimum at $ x=9 $), then from $ 9 $ to $ \infty $ it's increasing.
    So the decreasing intervals are $ (-8, -5) $, $ (-2, 3) $, $ (7, 9) $. Wait, but the options are $ (-\infty, -8) $, $ (-8, -5) $, $ (-8, -2) $, $ (-2, 3) $, $ (7, 9) $, $ (9, \infty) $. So:
        - $ (-8, -5) $: Decreasing (correct)
        - $ (-2, 3) $: Decreasing (correct)
        - $ (7, 9) $: Decreasing (correct)
        - $ (-\infty, -8) $: Increasing (incorrect)
        - $ (-8, -2) $: No, because from $ -5 $ to $ -2 $ it's increasing, so $ (-8, -2) $ has a decreasing part ($ -8 $ to $ -5 $) and an increasing part ($ -5 $ to $ -2 $), so the interval $ (-8, -2) $ is not entirely decreasing.
        - $ (9, \infty) $: Increasing (incorrect)

## Part (b)
### Step 1: Recall the definition of a local minimum
A local minimum of a function $ f(x) $ at a point $ x = c $ is a value such that $ f(c) $ is less than or equal to the values of $ f(x) $ for $ x $ near $ c $. Geometrically, it's a "valley" in the graph.

### Step 2: Identify local minima from the graph
Looking at the graph, the points where the graph has a local minimum (the bottom of the valleys) are at $ x=-5 $, $ x = 3 $, and $ x=9 $. We can see that at these $ x $ - values, the function changes from decreasing to increasing (which is the characteristic of a local minimum: the function decreases before the point and increases after the point).

# Answer:
(a) The intervals where the function is decreasing are $ (-8, -5) $, $ (-2, 3) $, $ (7, 9) $. So the correct options are $ \boldsymbol{(-8, -5)} $, $ \boldsymbol{(-2, 3)} $, $ \boldsymbol{(7, 9)} $.

(b) The $ x $ - values where the function has local minima are $ -5, 3, 9 $. So the answer is $ \boldsymbol{-5, 3, 9} $.

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Recall the definition of a decreasing function

Step 2: Analyze each interval

Part (b)

Step 1: Recall the definition of a local minimum

Step 2: Identify local minima from the graph

Answer: