The critical points of the function f(x)=((x−2)^2/3)(2x+1) are

To find the critical points of the function \( f(x) = \left( (x - 2)^{2/3} \right)(2x + 1) \), we need to follow these steps:

### 1. Differentiate the function \( f(x) \)
Given:
\[
f(x) = \left( (x - 2)^{2/3} \right)(2x + 1)
\]

We will use the product rule to differentiate \( f(x) \), where:
\[
u(x) = (x - 2)^{2/3}, \quad v(x) = 2x + 1
\]
The product rule states:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]

First, we calculate the derivatives \( u'(x) \) and \( v'(x) \):
- For \( u(x) = (x - 2)^{2/3} \):
  \[
  u'(x) = \frac{2}{3}(x - 2)^{-1/3}
  \]
- For \( v(x) = 2x + 1 \):
  \[
  v'(x) = 2
  \]

Now, applying the product rule:
\[
f'(x) = \frac{2}{3}(x - 2)^{-1/3}(2x + 1) + 2(x - 2)^{2/3}
\]

### 2. Simplify the derivative \( f'(x) \)
\[
f'(x) = \frac{2(2x + 1)}{3(x - 2)^{1/3}} + 2(x - 2)^{2/3}
\]

### 3. Find the critical points
Critical points occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined.

#### **a. Check where \( f'(x) = 0 \)**
Set the derivative equal to zero:
\[
\frac{2(2x + 1)}{3(x - 2)^{1/3}} + 2(x - 2)^{2/3} = 0
\]

To solve this, multiply through by \( 3(x - 2)^{1/3} \) to eliminate the fraction:
\[
2(2x + 1) + 6(x - 2) = 0
\]
Simplifying further:
\[
4x + 2 + 6x - 12 = 0
\]
\[
10x - 10 = 0
\]
\[
x = 1
\]

#### **b. Check where \( f'(x) \) is undefined**
The derivative \( f'(x) \) is undefined where \( (x - 2)^{1/3} = 0 \), which occurs at \( x = 2 \).

### 4. Conclusion
The critical points of the function \( f(x) = \left( (x - 2)^{2/3} \right)(2x + 1) \) are \( x = 1 \) and \( x = 2 \).

Would you like to explore further how these critical points affect the graph of the function or need any other details?

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Here are five follow-up questions to consider:

1. How can you determine whether each critical point is a maximum, minimum, or neither?
2. What is the significance of the point \( x = 2 \) where the derivative is undefined?
3. How does the behavior of the function change near each critical point?
4. Could there be any inflection points in this function, and how would you find them?
5. What role does the exponent \( 2/3 \) play in the nature of the critical points?

**Tip:** When the derivative is undefined at a point, it often indicates a cusp or a sharp change in the function, which is important to consider in the graph's overall shape.

Learn how to find critical points of the function f(x) = ((x - 2)^2/3)(2x + 1) using calculus. Understand the product rule for differentiation and how it applies in this context. Suitable for advanced high school students or college-level calculus students.

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