Solved on Dec 13, 2023

Find the equation of a function f(x)f(x) whose range decreases in 1<x<1-1<x<1 and increases in x>1x>1 and x<1x<-1. A. f(x)=x22x+1f(x)=x^{2}-2x+1 B. f(x)=x2+2x+1f(x)=x^{2}+2x+1 C. f(x)=x3+3x+5f(x)=x^{3}+3x+5 D. f(x)=x33x+5f(x)=x^{3}-3x+5

STEP 1

Assumptions
1. The function f(x)f(x) decreases in the interval 1<x<1-1 < x < 1.
2. The function f(x)f(x) increases in the intervals x>1x > 1 and x<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).

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)=x22x+1f(x) = x^2 - 2x + 1.

STEP 5

Calculate the first derivative of f(x)f(x) with respect to xx.
f(x)=ddx(x22x+1)f'(x) = \frac{d}{dx}(x^2 - 2x + 1)

STEP 6

Apply the power rule and the constant rule for differentiation.
f(x)=2x2f'(x) = 2x - 2

STEP 7

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

STEP 8

For x>1x > 1, f(x)=2x2f'(x) = 2x - 2 is positive, which means f(x)f(x) is increasing, which is consistent with the given condition.

STEP 9

For 1<x<1-1 < x < 1, f(x)=2x2f'(x) = 2x - 2 is negative, which means f(x)f(x) is decreasing, which is also consistent with the given condition.

STEP 10

For x<1x < -1, f(x)=2x2f'(x) = 2x - 2 is negative, which means f(x)f(x) is decreasing, which is not consistent with the given condition that f(x)f(x) should be increasing.

STEP 11

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

STEP 12

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

STEP 13

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

STEP 14

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

STEP 15

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

STEP 16

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

STEP 17

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

STEP 18

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

STEP 19

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

STEP 20

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

STEP 21

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

STEP 22

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

STEP 23

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

STEP 24

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

STEP 25

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

STEP 26

Move on to option D: f(x)=x33x+5f(x) = x^3 - 3x + 5.

STEP 27

Calculate the first derivative of f(x)f(x) with respect to xx.
f(x)=ddx(x33x+5)f'(x) = \frac{d}{dx}(x^3 - 3x + 5)

STEP 28

Apply the power rule and the constant rule for differentiation.
f(x)=3x23f'(x) = 3x^2 - 3

STEP 29

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

STEP 30

For x>1x > 1 and x<1x < -1, f(x)=3x23f'(x) = 3x^2 - 3 is positive, which means f(x)f(x) is increasing, which is consistent with the given conditions.

STEP 31

For 1<x<1-1 < x < 1, f(x)=3x23f'(x) = 3x^2 - 3 can be negative, which means 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) because it satisfies the conditions that the range of f(x)f(x) decreases for 1<x<1-1 < x < 1 and increases for x>1x > 1 and x<1x < -1.
The correct answer is D. f(x)=x33x+5f(x) = x^3 - 3x + 5.

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