Math / Calculus

QuestionWe want to find the relative extrema of $f(x)$ . Here is the graph of its derivative $f^{\prime}(x)$ .
Use the graph of $f^{\prime}(x)$ to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of $f(x)$ .

Studdy Solution

STEP 1

What is this asking? Where does our original function $f(x)$ have maximums, minimums, and plateaus, given the graph of its derivative $f'(x)$ ? Watch out! Don't mix up $f(x)$ and $f'(x)$ !
We're looking for properties of $f(x)$ , but we're given $f'(x)$ .

STEP 2

1. Analyze the Derivative's Sign
2. Identify Relative Extrema
3. Identify Horizontal Points of Inflection

STEP 3

Let's look at where $f'(x)$ is positive, negative, or zero. Why? Because the sign of the derivative tells us whether $f(x)$ is increasing or decreasing!

STEP 4

$f'(x)$ is positive before $x = -2$ .
This means $f(x)$ is increasing up to that point.
Then, $f'(x)$ becomes negative.
So $f(x)$ switches to decreasing after $x = -2$ .

STEP 5

$f'(x)$ is negative between $x = -2$ and $x = 3$ .
This means $f(x)$ is decreasing in this interval.

STEP 6

$f'(x)$ is positive between $x = 3$ and $x = 5$ .
This means $f(x)$ is increasing in this interval.

STEP 7

$f'(x)$ is negative after $x = 5$ .
This means $f(x)$ is decreasing after this point.

STEP 8

A relative maximum occurs when $f(x)$ stops increasing and starts decreasing.
This happens when $f'(x)$ changes from positive to negative.

STEP 9

Looking back at our analysis, this happens at $x = \mathbf{-2}$ .
So, we have a relative maximum there!

STEP 10

A relative minimum occurs when $f(x)$ stops decreasing and starts increasing.
This happens when $f'(x)$ changes from negative to positive.

STEP 11

This happens at $x = \mathbf{3}$ !
So, we have a relative minimum there!

STEP 12

A horizontal point of inflection occurs when $f'(x)$ is zero and the sign of $f'(x)$ doesn't change.
This means $f(x)$ momentarily flattens out but continues its current trend (increasing or decreasing).

STEP 13

At $x = \mathbf{5}$ , $f'(x)$ is zero, but it goes from positive to negative.
This isn't a horizontal point of inflection, it's a relative maximum (which we already identified!).

STEP 14

$f(x)$ has a relative maximum at $x = \mathbf{-2}$ , a relative minimum at $x = \mathbf{3}$ , and no horizontal points of inflection.

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QuestionWe want to find the relative extrema of $f(x)$ . Here is the graph of its derivative $f^{\prime}(x)$ .
Use the graph of $f^{\prime}(x)$ to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of $f(x)$ .

Studdy Solution

STEP 1

What is this asking? Where does our original function $f(x)$ have maximums, minimums, and plateaus, given the graph of its derivative $f'(x)$ ? Watch out! Don't mix up $f(x)$ and $f'(x)$ !
We're looking for properties of $f(x)$ , but we're given $f'(x)$ .

STEP 2

1. Analyze the Derivative's Sign
2. Identify Relative Extrema
3. Identify Horizontal Points of Inflection

STEP 3

Let's look at where $f'(x)$ is positive, negative, or zero. Why? Because the sign of the derivative tells us whether $f(x)$ is increasing or decreasing!

STEP 4

$f'(x)$ is positive before $x = -2$ .
This means $f(x)$ is increasing up to that point.
Then, $f'(x)$ becomes negative.
So $f(x)$ switches to decreasing after $x = -2$ .

STEP 5

$f'(x)$ is negative between $x = -2$ and $x = 3$ .
This means $f(x)$ is decreasing in this interval.

STEP 6

$f'(x)$ is positive between $x = 3$ and $x = 5$ .
This means $f(x)$ is increasing in this interval.

STEP 7

$f'(x)$ is negative after $x = 5$ .
This means $f(x)$ is decreasing after this point.

STEP 8

A relative maximum occurs when $f(x)$ stops increasing and starts decreasing.
This happens when $f'(x)$ changes from positive to negative.

STEP 9

Looking back at our analysis, this happens at $x = \mathbf{-2}$ .
So, we have a relative maximum there!

STEP 10

A relative minimum occurs when $f(x)$ stops decreasing and starts increasing.
This happens when $f'(x)$ changes from negative to positive.

STEP 11

This happens at $x = \mathbf{3}$ !
So, we have a relative minimum there!

STEP 12

A horizontal point of inflection occurs when $f'(x)$ is zero and the sign of $f'(x)$ doesn't change.
This means $f(x)$ momentarily flattens out but continues its current trend (increasing or decreasing).

STEP 13

At $x = \mathbf{5}$ , $f'(x)$ is zero, but it goes from positive to negative.
This isn't a horizontal point of inflection, it's a relative maximum (which we already identified!).

STEP 14

$f(x)$ has a relative maximum at $x = \mathbf{-2}$ , a relative minimum at $x = \mathbf{3}$ , and no horizontal points of inflection.

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QuestionWe want to find the relative extrema of f(x)f(x)f(x). Here is the graph of its derivative f′(x)f^{\prime}(x)f′(x).Use the graph of f′(x)f^{\prime}(x)f′(x) to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of f(x)f(x)f(x).

Studdy Solution

STEP 1

STEP 2

1. Analyze the Derivative's Sign2. Identify Relative Extrema3. Identify Horizontal Points of Inflection

STEP 3

Let's **look** at where f′(x)f'(x)f′(x) is positive, negative, or zero. *Why*? Because the sign of the derivative tells us whether f(x)f(x)f(x) is increasing or decreasing!

STEP 4

f′(x)f'(x)f′(x) is positive before x=−2x = -2x=−2.This means f(x)f(x)f(x) is **increasing** up to that point.Then, f′(x)f'(x)f′(x) becomes negative.So f(x)f(x)f(x) switches to **decreasing** after x=−2x = -2x=−2.

STEP 5

f′(x)f'(x)f′(x) is negative between x=−2x = -2x=−2 and x=3x = 3x=3.This means f(x)f(x)f(x) is **decreasing** in this interval.

STEP 6

f′(x)f'(x)f′(x) is positive between x=3x = 3x=3 and x=5x = 5x=5.This means f(x)f(x)f(x) is **increasing** in this interval.

STEP 7

f′(x)f'(x)f′(x) is negative after x=5x = 5x=5.This means f(x)f(x)f(x) is **decreasing** after this point.

STEP 8

A **relative maximum** occurs when f(x)f(x)f(x) *stops* increasing and *starts* decreasing.This happens when f′(x)f'(x)f′(x) changes from positive to negative.

STEP 9

Looking back at our analysis, this happens at x=−2x = \mathbf{-2}x=−2.So, we have a relative maximum there!

STEP 10

A **relative minimum** occurs when f(x)f(x)f(x) *stops* decreasing and *starts* increasing.This happens when f′(x)f'(x)f′(x) changes from negative to positive.

STEP 11

This happens at x=3x = \mathbf{3}x=3!So, we have a relative minimum there!

STEP 12

A **horizontal point of inflection** occurs when f′(x)f'(x)f′(x) is zero *and* the sign of f′(x)f'(x)f′(x) *doesn't* change.This means f(x)f(x)f(x) momentarily flattens out but continues its current trend (increasing or decreasing).

STEP 13

At x=5x = \mathbf{5}x=5, f′(x)f'(x)f′(x) is zero, but it goes from positive to negative.This isn't a horizontal point of inflection, it's a relative maximum (which we already identified!).

STEP 14

f(x)f(x)f(x) has a **relative maximum** at x=−2x = \mathbf{-2}x=−2, a **relative minimum** at x=3x = \mathbf{3}x=3, and *no* horizontal points of inflection.

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QuestionWe want to find the relative extrema of $f(x)$ . Here is the graph of its derivative $f^{\prime}(x)$ .
Use the graph of $f^{\prime}(x)$ to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of $f(x)$ .

1. Analyze the Derivative's Sign
2. Identify Relative Extrema
3. Identify Horizontal Points of Inflection

Let's look at where $f'(x)$ is positive, negative, or zero. Why? Because the sign of the derivative tells us whether $f(x)$ is increasing or decreasing!

$f'(x)$ is positive before $x = -2$ .
This means $f(x)$ is increasing up to that point.
Then, $f'(x)$ becomes negative.
So $f(x)$ switches to decreasing after $x = -2$ .

$f'(x)$ is negative between $x = -2$ and $x = 3$ .
This means $f(x)$ is decreasing in this interval.

$f'(x)$ is positive between $x = 3$ and $x = 5$ .
This means $f(x)$ is increasing in this interval.

$f'(x)$ is negative after $x = 5$ .
This means $f(x)$ is decreasing after this point.

A relative maximum occurs when $f(x)$ stops increasing and starts decreasing.
This happens when $f'(x)$ changes from positive to negative.

Looking back at our analysis, this happens at $x = \mathbf{-2}$ .
So, we have a relative maximum there!

A relative minimum occurs when $f(x)$ stops decreasing and starts increasing.
This happens when $f'(x)$ changes from negative to positive.

This happens at $x = \mathbf{3}$ !
So, we have a relative minimum there!

A horizontal point of inflection occurs when $f'(x)$ is zero and the sign of $f'(x)$ doesn't change.
This means $f(x)$ momentarily flattens out but continues its current trend (increasing or decreasing).

At $x = \mathbf{5}$ , $f'(x)$ is zero, but it goes from positive to negative.
This isn't a horizontal point of inflection, it's a relative maximum (which we already identified!).

$f(x)$ has a relative maximum at $x = \mathbf{-2}$ , a relative minimum at $x = \mathbf{3}$ , and no horizontal points of inflection.