Solved on Dec 13, 2023

Find the equation of a function $f(x)$ whose range decreases in $-1<x<1$ and increases in $x>1$ and $x<-1$ . A. $f(x)=x^{2}-2x+1$ B. $f(x)=x^{2}+2x+1$ C. $f(x)=x^{3}+3x+5$ D. $f(x)=x^{3}-3x+5$

STEP 1

Assumptions
1. The function $f(x)$ decreases in the interval $-1 < x < 1$ .
2. The function $f(x)$ increases in the intervals $x > 1$ and $x < -1$ .
3. We are looking for a function that satisfies these conditions.

STEP 2

To determine the intervals of increase and decrease for a function, we need to find its first derivative, $f'(x)$ .

STEP 3

We will calculate the first derivative of each function option and analyze the sign of the derivative in the given intervals.

STEP 4

Start with option A: $f(x) = x^2 - 2x + 1$ .

STEP 5

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^2 - 2x + 1)$

STEP 6

Apply the power rule and the constant rule for differentiation.
$f'(x) = 2x - 2$

STEP 7

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

STEP 8

For $x > 1$ , $f'(x) = 2x - 2$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

STEP 9

For $-1 < x < 1$ , $f'(x) = 2x - 2$ is negative, which means $f(x)$ is decreasing, which is also consistent with the given condition.

STEP 10

For $x < -1$ , $f'(x) = 2x - 2$ is negative, which means $f(x)$ is decreasing, which is not consistent with the given condition that $f(x)$ should be increasing.

STEP 11

Therefore, option A cannot be the correct equation for $f(x)$ .

STEP 12

Move on to option B: $f(x) = x^2 + 2x + 1$ .

STEP 13

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^2 + 2x + 1)$

STEP 14

Apply the power rule and the constant rule for differentiation.
$f'(x) = 2x + 2$

STEP 15

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

STEP 16

For $x > 1$ , $f'(x) = 2x + 2$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

STEP 17

For $-1 < x < 1$ , $f'(x) = 2x + 2$ is positive, which means $f(x)$ is increasing, which is not consistent with the given condition that $f(x)$ should be decreasing.

STEP 18

Therefore, option B cannot be the correct equation for $f(x)$ .

STEP 19

Move on to option C: $f(x) = x^3 + 3x + 5$ .

STEP 20

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^3 + 3x + 5)$

STEP 21

Apply the power rule and the constant rule for differentiation.
$f'(x) = 3x^2 + 3$

STEP 22

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

STEP 23

For $x > 1$ , $f'(x) = 3x^2 + 3$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

STEP 24

For $-1 < x < 1$ , $f'(x) = 3x^2 + 3$ is positive, which means $f(x)$ is increasing, which is not consistent with the given condition that $f(x)$ should be decreasing.

STEP 25

Therefore, option C cannot be the correct equation for $f(x)$ .

STEP 26

Move on to option D: $f(x) = x^3 - 3x + 5$ .

STEP 27

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^3 - 3x + 5)$

STEP 28

Apply the power rule and the constant rule for differentiation.
$f'(x) = 3x^2 - 3$

STEP 29

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

STEP 30

For $x > 1$ and $x < -1$ , $f'(x) = 3x^2 - 3$ is positive, which means $f(x)$ is increasing, which is consistent with the given conditions.

STEP 31

For $-1 < x < 1$ , $f'(x) = 3x^2 - 3$ can be negative, which means $f(x)$ is decreasing, which is also consistent with the given conditions.

STEP 32

Therefore, option D is the correct equation for $f(x)$ because it satisfies the conditions that the range of $f(x)$ decreases for $-1 < x < 1$ and increases for $x > 1$ and $x < -1$ .
The correct answer is D. $f(x) = x^3 - 3x + 5$ .

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STEP 1

Assumptions1. The function f(x)f(x)f(x) decreases in the interval −1<x<1-1 < x < 1−1<x<1.2. The function f(x)f(x)f(x) increases in the intervals x>1x > 1x>1 and x<−1x < -1x<−1.3. We are looking for a function that satisfies these conditions.

STEP 2

To determine the intervals of increase and decrease for a function, we need to find its first derivative, f′(x)f'(x)f′(x).

STEP 3

We will calculate the first derivative of each function option and analyze the sign of the derivative in the given intervals.

STEP 4

Start with option A: f(x)=x2−2x+1f(x) = x^2 - 2x + 1f(x)=x2−2x+1.

STEP 5

Calculate the first derivative of f(x)f(x)f(x) with respect to xxx.f′(x)=ddx(x2−2x+1)f'(x) = \frac{d}{dx}(x^2 - 2x + 1)f′(x)=dxd​(x2−2x+1)

STEP 6

Apply the power rule and the constant rule for differentiation.f′(x)=2x−2f'(x) = 2x - 2f′(x)=2x−2

STEP 7

Analyze the sign of f′(x)f'(x)f′(x) in the intervals x>1x > 1x>1, −1<x<1-1 < x < 1−1<x<1, and x<−1x < -1x<−1.

STEP 8

For x>1x > 1x>1, f′(x)=2x−2f'(x) = 2x - 2f′(x)=2x−2 is positive, which means f(x)f(x)f(x) is increasing, which is consistent with the given condition.

STEP 9

For −1<x<1-1 < x < 1−1<x<1, f′(x)=2x−2f'(x) = 2x - 2f′(x)=2x−2 is negative, which means f(x)f(x)f(x) is decreasing, which is also consistent with the given condition.

STEP 10

For x<−1x < -1x<−1, f′(x)=2x−2f'(x) = 2x - 2f′(x)=2x−2 is negative, which means f(x)f(x)f(x) is decreasing, which is not consistent with the given condition that f(x)f(x)f(x) should be increasing.

STEP 11

Therefore, option A cannot be the correct equation for f(x)f(x)f(x).

STEP 12

Move on to option B: f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1.

STEP 13

Calculate the first derivative of f(x)f(x)f(x) with respect to xxx.f′(x)=ddx(x2+2x+1)f'(x) = \frac{d}{dx}(x^2 + 2x + 1)f′(x)=dxd​(x2+2x+1)

STEP 14

Apply the power rule and the constant rule for differentiation.f′(x)=2x+2f'(x) = 2x + 2f′(x)=2x+2

STEP 15

Analyze the sign of f′(x)f'(x)f′(x) in the intervals x>1x > 1x>1, −1<x<1-1 < x < 1−1<x<1, and x<−1x < -1x<−1.

STEP 16

For x>1x > 1x>1, f′(x)=2x+2f'(x) = 2x + 2f′(x)=2x+2 is positive, which means f(x)f(x)f(x) is increasing, which is consistent with the given condition.

STEP 17

For −1<x<1-1 < x < 1−1<x<1, f′(x)=2x+2f'(x) = 2x + 2f′(x)=2x+2 is positive, which means f(x)f(x)f(x) is increasing, which is not consistent with the given condition that f(x)f(x)f(x) should be decreasing.

STEP 18

Therefore, option B cannot be the correct equation for f(x)f(x)f(x).

STEP 19

Move on to option C: f(x)=x3+3x+5f(x) = x^3 + 3x + 5f(x)=x3+3x+5.

STEP 20

Calculate the first derivative of f(x)f(x)f(x) with respect to xxx.f′(x)=ddx(x3+3x+5)f'(x) = \frac{d}{dx}(x^3 + 3x + 5)f′(x)=dxd​(x3+3x+5)

STEP 21

Apply the power rule and the constant rule for differentiation.f′(x)=3x2+3f'(x) = 3x^2 + 3f′(x)=3x2+3

STEP 22

Analyze the sign of f′(x)f'(x)f′(x) in the intervals x>1x > 1x>1, −1<x<1-1 < x < 1−1<x<1, and x<−1x < -1x<−1.

STEP 23

For x>1x > 1x>1, f′(x)=3x2+3f'(x) = 3x^2 + 3f′(x)=3x2+3 is positive, which means f(x)f(x)f(x) is increasing, which is consistent with the given condition.

STEP 24

For −1<x<1-1 < x < 1−1<x<1, f′(x)=3x2+3f'(x) = 3x^2 + 3f′(x)=3x2+3 is positive, which means f(x)f(x)f(x) is increasing, which is not consistent with the given condition that f(x)f(x)f(x) should be decreasing.

STEP 25

Therefore, option C cannot be the correct equation for f(x)f(x)f(x).

STEP 26

Move on to option D: f(x)=x3−3x+5f(x) = x^3 - 3x + 5f(x)=x3−3x+5.

STEP 27

Calculate the first derivative of f(x)f(x)f(x) with respect to xxx.f′(x)=ddx(x3−3x+5)f'(x) = \frac{d}{dx}(x^3 - 3x + 5)f′(x)=dxd​(x3−3x+5)

STEP 28

Apply the power rule and the constant rule for differentiation.f′(x)=3x2−3f'(x) = 3x^2 - 3f′(x)=3x2−3

STEP 29

Analyze the sign of f′(x)f'(x)f′(x) in the intervals x>1x > 1x>1, −1<x<1-1 < x < 1−1<x<1, and x<−1x < -1x<−1.

STEP 30

For x>1x > 1x>1 and x<−1x < -1x<−1, f′(x)=3x2−3f'(x) = 3x^2 - 3f′(x)=3x2−3 is positive, which means f(x)f(x)f(x) is increasing, which is consistent with the given conditions.

STEP 31

For −1<x<1-1 < x < 1−1<x<1, f′(x)=3x2−3f'(x) = 3x^2 - 3f′(x)=3x2−3 can be negative, which means f(x)f(x)f(x) is decreasing, which is also consistent with the given conditions.

STEP 32

Therefore, option D is the correct equation for f(x)f(x)f(x) because it satisfies the conditions that the range of f(x)f(x)f(x) decreases for −1<x<1-1 < x < 1−1<x<1 and increases for x>1x > 1x>1 and x<−1x < -1x<−1.The correct answer is D. f(x)=x3−3x+5f(x) = x^3 - 3x + 5f(x)=x3−3x+5.

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Assumptions
1. The function $f(x)$ decreases in the interval $-1 < x < 1$ .
2. The function $f(x)$ increases in the intervals $x > 1$ and $x < -1$ .
3. We are looking for a function that satisfies these conditions.

To determine the intervals of increase and decrease for a function, we need to find its first derivative, $f'(x)$ .

Start with option A: $f(x) = x^2 - 2x + 1$ .

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^2 - 2x + 1)$

Apply the power rule and the constant rule for differentiation.
$f'(x) = 2x - 2$

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

For $x > 1$ , $f'(x) = 2x - 2$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

For $-1 < x < 1$ , $f'(x) = 2x - 2$ is negative, which means $f(x)$ is decreasing, which is also consistent with the given condition.

For $x < -1$ , $f'(x) = 2x - 2$ is negative, which means $f(x)$ is decreasing, which is not consistent with the given condition that $f(x)$ should be increasing.

Therefore, option A cannot be the correct equation for $f(x)$ .

Move on to option B: $f(x) = x^2 + 2x + 1$ .

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^2 + 2x + 1)$

Apply the power rule and the constant rule for differentiation.
$f'(x) = 2x + 2$

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

For $x > 1$ , $f'(x) = 2x + 2$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

For $-1 < x < 1$ , $f'(x) = 2x + 2$ is positive, which means $f(x)$ is increasing, which is not consistent with the given condition that $f(x)$ should be decreasing.

Therefore, option B cannot be the correct equation for $f(x)$ .

Move on to option C: $f(x) = x^3 + 3x + 5$ .

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^3 + 3x + 5)$

Apply the power rule and the constant rule for differentiation.
$f'(x) = 3x^2 + 3$

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

For $x > 1$ , $f'(x) = 3x^2 + 3$ is positive, which means $f(x)$ is increasing, which is consistent with the given condition.

For $-1 < x < 1$ , $f'(x) = 3x^2 + 3$ is positive, which means $f(x)$ is increasing, which is not consistent with the given condition that $f(x)$ should be decreasing.

Therefore, option C cannot be the correct equation for $f(x)$ .

Move on to option D: $f(x) = x^3 - 3x + 5$ .

Calculate the first derivative of $f(x)$ with respect to $x$ .
$f'(x) = \frac{d}{dx}(x^3 - 3x + 5)$

Apply the power rule and the constant rule for differentiation.
$f'(x) = 3x^2 - 3$

Analyze the sign of $f'(x)$ in the intervals $x > 1$ , $-1 < x < 1$ , and $x < -1$ .

For $x > 1$ and $x < -1$ , $f'(x) = 3x^2 - 3$ is positive, which means $f(x)$ is increasing, which is consistent with the given conditions.

For $-1 < x < 1$ , $f'(x) = 3x^2 - 3$ can be negative, which means $f(x)$ is decreasing, which is also consistent with the given conditions.

Therefore, option D is the correct equation for $f(x)$ because it satisfies the conditions that the range of $f(x)$ decreases for $-1 < x < 1$ and increases for $x > 1$ and $x < -1$ .
The correct answer is D. $f(x) = x^3 - 3x + 5$ .