The graph of f(x) = 3x(x - 2)² has a local minimum value at :
 A. the point (2 ; 0)
 B. the point (0 ; 2)
 C. x = 0
 D. x = 2

To find the local minimum value of the function \( f(x) = 3x(x - 2)^2 \), we need to follow these steps:

1. **Find the first derivative of the function** to determine the critical points:
   \[
   f(x) = 3x(x - 2)^2
   \]
   Apply the product rule and chain rule to differentiate:
   \[
   f'(x) = 3[(x - 2)^2 + 2x(x - 2)]
   \]
   Simplify the derivative:
   \[
   f'(x) = 3[(x - 2)(x - 2) + 2x(x - 2)]
   \]
   Factor the expression:
   \[
   f'(x) = 3(x - 2)[(x - 2) + 2x] = 3(x - 2)(3x - 2)
   \]

2. **Set the first derivative equal to zero** to find the critical points:
   \[
   3(x - 2)(3x + 2) = 0
   \]
   This gives us two critical points:
   \[
   x = 2 \quad \text{and} \quad x = 0
   \]

3. **Determine the nature of these critical points** using the second derivative test or by analyzing the sign of the first derivative around these points.

   Let's substitute \( x = 0 \) and \( x = 2 \) back into the function \( f(x) \) to check their corresponding values:
   \[
   f(0) = 3(0)(0 - 2)^2 = 0
   \]
   \[
   f(2) = 3(2)(2 - 2)^2 = 0
   \]

   Since both yield \( f(x) = 0 \), check the behavior of \( f(x) \) around \( x = 0 \) and \( x = 2 \):
   - As \( x \) increases past 0, the function decreases and then increases, indicating \( x = 0 \) is a maximum.
   - As \( x \) increases past 2, the function decreases before increasing again, indicating \( x = 2 \) is a minimum.

Therefore, the correct answer is **A. the point (2, 0)**.

Would you like more details or have any questions? Here are some related questions:

1. How do you calculate the first derivative of a function using the product rule?
2. What is the second derivative test and when should it be used?
3. How do you determine if a critical point is a local minimum, maximum, or neither?
4. Can there be more than one local minimum or maximum for a function?
5. What is the significance of the sign of the derivative in determining the behavior of a function?

**Tip:** Always double-check your critical points by analyzing the sign of the first derivative around them to confirm their nature (minimum, maximum, or inflection point).

Learn how to find the local minimum value of the function f(x) = 3x(x - 2)^2 using calculus and the first derivative test. Understand the concept of critical points and local extrema in calculus. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation