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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