Questions: 1a. On what open intervals is f(x) increasing? -3<x<3 1b. On what open intervals is f(x) concave down? 0<x<0.5 1c. On what interval is the rate of change of f(x) negative and decreasing? -6<x<-3

Transcript text: 1. 1a. On what open intervals is $f(x)$ increasing? \[ -3

Solution

Step 1: Identify intervals where $ f(x) $ is increasing

To determine where $ f(x) $ is increasing, look for intervals where the graph is moving upwards. This occurs between the local minima and maxima.

From the graph, $ f(x) $ is increasing on the intervals $ (-\infty, -1) $ and $ (0, 2) $.

Step 2: Identify intervals where $ f(x) $ is concave down

To determine where $ f(x) $ is concave down, look for intervals where the graph is curving downwards. This occurs between points of inflection.

From the graph, $ f(x) $ is concave down on the intervals $ (-1, 1) $ and $ (2, \infty) $.

Step 3: Identify intervals where the rate of change of $ f(x) $ is negative

To determine where the rate of change of $ f(x) $ is negative, look for intervals where the graph is decreasing.

From the graph, $ f(x) $ is decreasing on the intervals $ (-1, 0) $ and $ (2, \infty) $.

1a. $ f(x) $ is increasing on the intervals $ (-\infty, -1) $ and $ (0, 2) $.

1b. $ f(x) $ is concave down on the intervals $ (-1, 1) $ and $ (2, \infty) $.

1c. The rate of change of $ f(x) $ is negative on the intervals $ (-1, 0) $ and $ (2, \infty) $.

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