https://file.aiquickdraw.com/mathbot/file/172566370445181kpqpq3.jpg

The image contains four functions, and the task is to find the interval of increase or decrease for each function. Let's go through them one by one.

1. **\( f(x) = x^2 - 12 \):**

   - **Derivative**: \( f'(x) = 2x \).
   - Set \( f'(x) = 0 \) to find critical points: \( 2x = 0 \) gives \( x = 0 \).
   - Check the sign of \( f'(x) \) around the critical point:
     - For \( x < 0 \), \( f'(x) < 0 \), so \( f(x) \) is decreasing.
     - For \( x > 0 \), \( f'(x) > 0 \), so \( f(x) \) is increasing.
   - **Intervals**:
     - Decreasing: \( (-\infty, 0) \)
     - Increasing: \( (0, \infty) \)

2. **\( g(x) = \frac{1}{2}x + 1 \):**

   - **Derivative**: \( g'(x) = \frac{1}{2} \), which is constant and positive.
   - Since the derivative is positive everywhere, the function is always increasing.
   - **Intervals**:
     - Increasing: \( (-\infty, \infty) \)

3. **\( h(x) = -\frac{x^2}{2} \):**

   - **Derivative**: \( h'(x) = -x \).
   - Set \( h'(x) = 0 \) to find critical points: \( -x = 0 \) gives \( x = 0 \).
   - Check the sign of \( h'(x) \) around the critical point:
     - For \( x < 0 \), \( h'(x) > 0 \), so \( h(x) \) is increasing.
     - For \( x > 0 \), \( h'(x) < 0 \), so \( h(x) \) is decreasing.
   - **Intervals**:
     - Increasing: \( (-\infty, 0) \)
     - Decreasing: \( (0, \infty) \)

4. **\( j(x) = \frac{1}{x} \):**

   - **Derivative**: \( j'(x) = -\frac{1}{x^2} \).
   - The derivative \( j'(x) \) is always negative for all \( x \neq 0 \), meaning the function is always decreasing.
   - **Intervals**:
     - Decreasing: \( (-\infty, 0) \cup (0, \infty) \)

Would you like to explore more details or have any questions about these intervals? Here are five relative questions to deepen your understanding:

1. How do critical points help in determining intervals of increase or decrease?
2. Why is the sign of the derivative significant in determining whether a function is increasing or decreasing?
3. How does the degree of a polynomial affect its behavior and intervals of increase/decrease?
4. Can you graph these functions to visualize their increasing and decreasing intervals?
5. How would the intervals of increase or decrease change if the coefficients in the functions were different?

**Tip:** Always check the behavior of the function at critical points and endpoints to accurately determine the intervals of increase or decrease.

Find the intervals of increase or decrease for the given functions: f(x) = x^2 - 12, g(x) = 0.5x + 1, h(x) = -x^2/2, j(x) = 1/x.

Explore how to determine the intervals of increase and decrease for various functions including quadratic and rational functions. This analysis involves derivatives and critical points, key in identifying function behavior over its domain.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation