To find the intervals on which the function \( f(x) = 3x \) is increasing, decreasing, or constant, we analyze the behavior of the function.

### 1. **Derivative of the function:**
The first step is to find the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(3x) = 3
\]
The derivative \( f'(x) = 3 \) is a constant positive value.

### 2. **Analysis of the derivative:**
- Since \( f'(x) = 3 > 0 \) for all \( x \), the function is **increasing** on the entire real line.
- The function \( f(x) = 3x \) does not have any points where it is decreasing or constant because the derivative is never negative or zero.

### 3. **Conclusion:**
The function \( f(x) = 3x \) is increasing on the entire real line \( (-\infty, \infty) \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What does the slope of the derivative tell us about the function?
2. How does the sign of the derivative determine intervals of increase or decrease?
3. Can you find intervals of increase and decrease for a quadratic function?
4. What is the geometric interpretation of a linear function like \( f(x) = 3x \)?
5. How do you determine critical points and what do they signify?

**Tip:** For a linear function like \( f(x) = 3x \), the behavior (increasing or decreasing) is determined entirely by the sign of the slope (coefficient of \( x \)).

Discover how to find intervals of increase, decrease, and constancy for the function f(x) = 3x. Understand the behavior of linear functions using calculus concepts and derivative analysis. Suitable for advanced high school students and anyone interested in calculus and function analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation