Question

5.
a) on what intervals of x is f(x) positive?
b) on what intervals of x does f(x) have a positive slope?
c) on what intervals of x is f(x) increasing?

6.
a) on what intervals of x is f(x) decreasing?
b) on what intervals of x does f(x) have a positive slope?
c) on what intervals of x is f(x) positive?
d) on what intervals of x does f(x) have a zero slope?

7.
a) on what intervals of x is f(x) negative?
b) on what intervals of x is f(x) decreasing?
c) on what interval of x does f(x) have negative slopes?

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Explanation:

Problem 5

Step1: Identify positive f(x)

$f(x) > 0$ when the graph is above the x-axis. The graph is above the x-axis for all $x$ in its domain.

Step2: Identify positive slope

A positive slope means the graph rises from left to right. This is true for all $x$ in the domain.

Step3: Identify increasing f(x)

A function is increasing when it rises left to right, same as positive slope.

Step1: Identify decreasing f(x)

$f(x)$ decreases when the graph falls left to right: $x > 2$.

Step2: Identify positive slope

Positive slope means the graph rises left to right: $x < -3$.

Step3: Identify positive f(x)

$f(x) > 0$ when the graph is above the x-axis: $x < 3$.

Step4: Identify zero slope

Zero slope means the graph is horizontal: $-3 < x < 2$.

Step1: Identify negative f(x)

$f(x) < 0$ when the graph is below the x-axis: $-2 < x < 0$ and $2 < x < 4$.

Step2: Identify decreasing f(x)

$f(x)$ decreases when the graph falls left to right: $-4 < x < -2$, $0 < x < 2$, $4 < x < 6$.

Step3: Identify negative slope

Negative slope means the graph falls left to right, same as decreasing intervals.

Answer:

a) $(-\infty, \infty)$
b) $(-\infty, \infty)$
c) $(-\infty, \infty)$

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Problem 6