Math / Calculus

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The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-2 x-5 \sin (2 x-2)$ for $-2<x<4$ . Find all intervals in the given domain where the function $f$ is increasing. You may use a calculator and round all values to 3 decimal places. Answer Attempt 2 out of 3

Studdy Solution

STEP 1

1. The function $f$ is differentiable on the interval $-2 < x < 4$ .
2. The derivative of the function $f$ is given by $f^{\prime}(x) = x^2 - 2x - 5 \sin(2x - 2)$ .
3. We need to find intervals where $f$ is increasing, which occurs where $f^{\prime}(x) > 0$ .

STEP 2

1. Determine where $f^{\prime}(x) > 0$ .
2. Solve the inequality to find intervals.
3. Verify intervals using a calculator.

STEP 3

To find where $f$ is increasing, solve the inequality $f^{\prime}(x) > 0$ .
This means solving: $x^2 - 2x - 5 \sin(2x - 2) > 0$

STEP 4

Use a calculator to evaluate $f^{\prime}(x)$ at several points in the interval $-2 < x < 4$ to approximate where the inequality holds.

STEP 5

Identify intervals where the evaluated values of $f^{\prime}(x)$ are positive.

STEP 6

Using the calculator, find the critical points of $f^{\prime}(x)$ by setting $f^{\prime}(x) = 0$ and solving for $x$ .

STEP 7

Verify the intervals by checking the sign of $f^{\prime}(x)$ around the critical points and within the domain $-2 < x < 4$ .
The intervals where $f$ is increasing are approximately:
$(a, b)$ and $(c, d)$
where $a, b, c,$ and $d$ are the critical points found using the calculator.

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Studdy Solution

STEP 1

1. The function $f$ is differentiable on the interval $-2 < x < 4$ .
2. The derivative of the function $f$ is given by $f^{\prime}(x) = x^2 - 2x - 5 \sin(2x - 2)$ .
3. We need to find intervals where $f$ is increasing, which occurs where $f^{\prime}(x) > 0$ .

STEP 2

1. Determine where $f^{\prime}(x) > 0$ .
2. Solve the inequality to find intervals.
3. Verify intervals using a calculator.

STEP 3

To find where $f$ is increasing, solve the inequality $f^{\prime}(x) > 0$ .
This means solving: $x^2 - 2x - 5 \sin(2x - 2) > 0$

STEP 4

Use a calculator to evaluate $f^{\prime}(x)$ at several points in the interval $-2 < x < 4$ to approximate where the inequality holds.

STEP 5

Identify intervals where the evaluated values of $f^{\prime}(x)$ are positive.

STEP 6

Using the calculator, find the critical points of $f^{\prime}(x)$ by setting $f^{\prime}(x) = 0$ and solving for $x$ .

STEP 7

Verify the intervals by checking the sign of $f^{\prime}(x)$ around the critical points and within the domain $-2 < x < 4$ .
The intervals where $f$ is increasing are approximately:
$(a, b)$ and $(c, d)$
where $a, b, c,$ and $d$ are the critical points found using the calculator.

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Studdy Solution

STEP 1

STEP 2

1. Determine where f′(x)>0 f^{\prime}(x) > 0 f′(x)>0.2. Solve the inequality to find intervals.3. Verify intervals using a calculator.

STEP 3

To find where f f f is increasing, solve the inequality f′(x)>0 f^{\prime}(x) > 0 f′(x)>0.This means solving: x2−2x−5sin⁡(2x−2)>0 x^2 - 2x - 5 \sin(2x - 2) > 0 x2−2x−5sin(2x−2)>0

STEP 4

Use a calculator to evaluate f′(x) f^{\prime}(x) f′(x) at several points in the interval −2<x<4-2 < x < 4−2<x<4 to approximate where the inequality holds.

STEP 5

Identify intervals where the evaluated values of f′(x) f^{\prime}(x) f′(x) are positive.

STEP 6

Using the calculator, find the critical points of f′(x) f^{\prime}(x) f′(x) by setting f′(x)=0 f^{\prime}(x) = 0 f′(x)=0 and solving for x x x.

STEP 7

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1. Determine where $f^{\prime}(x) > 0$ .
2. Solve the inequality to find intervals.
3. Verify intervals using a calculator.

To find where $f$ is increasing, solve the inequality $f^{\prime}(x) > 0$ .
This means solving: $x^2 - 2x - 5 \sin(2x - 2) > 0$

Use a calculator to evaluate $f^{\prime}(x)$ at several points in the interval $-2 < x < 4$ to approximate where the inequality holds.

Identify intervals where the evaluated values of $f^{\prime}(x)$ are positive.

Using the calculator, find the critical points of $f^{\prime}(x)$ by setting $f^{\prime}(x) = 0$ and solving for $x$ .