https://file.aiquickdraw.com/mathbot/file/17273138400261v17j2zf.jpg

Jawablah pertanyaan diatas lengkap dengan caranya

The problem asks for the intervals where both functions \( f(x) = 2x^3 - 9x^2 + 12x + 4 \) and \( g(x) = x^2 + 6x + 1 \) are increasing.

### Step 1: Identify the intervals of increasing functions
To determine where a function is increasing, we need to find where its derivative is positive. This means we need to calculate the first derivative of each function and analyze the sign of the derivative.

#### Derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(2x^3 - 9x^2 + 12x + 4) = 6x^2 - 18x + 12
\]
We now find the critical points by setting \( f'(x) = 0 \):
\[
6x^2 - 18x + 12 = 0
\]
Dividing the equation by 6:
\[
x^2 - 3x + 2 = 0
\]
Factoring the quadratic:
\[
(x - 1)(x - 2) = 0
\]
Thus, \( x = 1 \) and \( x = 2 \) are the critical points.

#### Derivative of \( g(x) \):
\[
g'(x) = \frac{d}{dx}(x^2 + 6x + 1) = 2x + 6
\]
Setting \( g'(x) = 0 \) to find the critical points:
\[
2x + 6 = 0
\]
\[
x = -3
\]

### Step 2: Analyze the sign of the derivatives
We now analyze the sign of the derivatives on different intervals:

#### For \( f'(x) = 6x^2 - 18x + 12 \):
The critical points are \( x = 1 \) and \( x = 2 \). Let's test the sign of \( f'(x) \) in the intervals \( (-\infty, 1) \), \( (1, 2) \), and \( (2, \infty) \):

- For \( x < 1 \) (e.g., \( x = 0 \)):
  \[
  f'(0) = 6(0)^2 - 18(0) + 12 = 12 \quad (\text{positive})
  \]
- For \( 1 < x < 2 \) (e.g., \( x = 1.5 \)):
  \[
  f'(1.5) = 6(1.5)^2 - 18(1.5) + 12 = -1.5 \quad (\text{negative})
  \]
- For \( x > 2 \) (e.g., \( x = 3 \)):
  \[
  f'(3) = 6(3)^2 - 18(3) + 12 = 30 \quad (\text{positive})
  \]

So, \( f(x) \) is increasing for \( x < 1 \) and \( x > 2 \).

#### For \( g'(x) = 2x + 6 \):
The critical point is \( x = -3 \). Let's test the sign of \( g'(x) \) in the intervals \( (-\infty, -3) \) and \( (-3, \infty) \):

- For \( x < -3 \) (e.g., \( x = -4 \)):
  \[
  g'(-4) = 2(-4) + 6 = -2 \quad (\text{negative})
  \]
- For \( x > -3 \) (e.g., \( x = 0 \)):
  \[
  g'(0) = 2(0) + 6 = 6 \quad (\text{positive})
  \]

So, \( g(x) \) is increasing for \( x > -3 \).

### Step 3: Determine where both functions are increasing
- \( f(x) \) is increasing for \( x < 1 \) and \( x > 2 \).
- \( g(x) \) is increasing for \( x > -3 \).

Thus, the interval where both functions are increasing is:
- \( (-3, 1) \) or \( x > 2 \).

This corresponds to option **C**: \( -3 < x < 1 \) or \( x > 2 \).

### Final Answer:
The correct answer is **C**.

Would you like further details on any step? Here are some related questions to expand your understanding:

1. How do you determine whether a function is increasing or decreasing from its derivative?
2. What are critical points, and why are they important in analyzing the behavior of functions?
3. How do you factor quadratics like \( x^2 - 3x + 2 \)?
4. What happens if the derivative of a function is zero at a point?
5. Can a function have intervals where it is neither increasing nor decreasing?

**Tip**: Always check the sign of the derivative in intervals divided by critical points to ensure you identify the correct intervals where the function is increasing or decreasing.

Fungsi f(x) = 2x^3 - 9x^2 + 12x + 4 dan g(x) = x^2 + 6x + 1 akan sama-sama naik pada interval…

Learn how to determine the intervals where the functions f(x) = 2x^3 - 9x^2 + 12x + 4 and g(x) = x^2 + 6x + 1 are both increasing. This solution explains how to use derivatives to find critical points and analyze the signs of the derivatives.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation