https://file.aiquickdraw.com/mathbot/file/1724482757895z8xoat9c.jpg

Find the interval of values of x for which the function is decreasing.

To find the interval where the function \( f(x) = 3x^2 - 6x + 2 \) is decreasing, we need to follow these steps:

1. **Find the derivative \( f'(x) \)**:
   The first derivative of the function will give us the rate of change of \( f(x) \) with respect to \( x \). The function is decreasing where this derivative is negative.

   \[
   f'(x) = \frac{d}{dx} \left( 3x^2 - 6x + 2 \right)
   \]

   \[
   f'(x) = 6x - 6
   \]

2. **Determine where \( f'(x) < 0 \)**:
   The function is decreasing where the derivative is less than zero.

   \[
   6x - 6 < 0
   \]

   \[
   6x < 6
   \]

   \[
   x < 1
   \]

Thus, the function \( f(x) = 3x^2 - 6x + 2 \) is decreasing on the interval \( (-\infty, 1) \).

Would you like any further details or have any questions?

Here are some related questions you might consider:
1. What is the interval where the function is increasing?
2. How do you find the critical points of a function?
3. What is the second derivative of \( f(x) \) and what does it indicate?
4. How do you determine the local minimum and maximum points of a function?
5. How can you find the vertex of the quadratic function \( f(x) = 3x^2 - 6x + 2 \)?

**Tip:** When analyzing the behavior of functions, always start by finding the derivative. The sign of the derivative helps determine if the function is increasing or decreasing.

Learn how to find the interval where the function f(x) = 3x^2 - 6x + 2 is decreasing with a step-by-step explanation. This problem involves calculus concepts such as derivatives and function behavior analysis. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation