Consider the function f(x) = 7 (x - 2)^{2/3}. For this function there are two important intervals: (-\infty, A) and (A,\infty) where A is a critical number.
Find A 

For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC).
(-\infty, A): 

(A,\infty): 


Let's start by finding the critical number \( A \) for the function \( f(x) = 7(x - 2)^{2/3} \). Critical numbers occur where the derivative is zero or undefined.

### Step 1: Differentiate the function

To find the critical points, we first compute the derivative of \( f(x) \).

Given:
\[
f(x) = 7(x - 2)^{2/3}
\]

Using the chain rule:
\[
f'(x) = 7 \cdot \frac{2}{3}(x - 2)^{-1/3} \cdot 1 = \frac{14}{3}(x - 2)^{-1/3}
\]

### Step 2: Find the critical point

Critical points occur where \( f'(x) = 0 \) or where the derivative is undefined.

- \( f'(x) = 0 \): For this to happen, the numerator of the derivative must be zero, but the numerator is a constant \( 14 \), so the derivative is never zero.
  
- Undefined: The derivative is undefined when \( (x - 2) = 0 \), or \( x = 2 \).

Thus, \( A = 2 \) is the critical number.

### Step 3: Determine where \( f(x) \) is increasing or decreasing

We now analyze the intervals \( (-\infty, 2) \) and \( (2, \infty) \) to determine whether \( f(x) \) is increasing or decreasing.

- On \( (-\infty, 2) \):
    - Choose a test point \( x = 1 \).
    - \( f'(1) = \frac{14}{3}(1 - 2)^{-1/3} = \frac{14}{3}(-1)^{-1/3} = -\frac{14}{3} \).
    - Since \( f'(1) < 0 \), \( f(x) \) is **decreasing** on \( (-\infty, 2) \).

- On \( (2, \infty) \):
    - Choose a test point \( x = 3 \).
    - \( f'(3) = \frac{14}{3}(3 - 2)^{-1/3} = \frac{14}{3}(1)^{-1/3} = \frac{14}{3} \).
    - Since \( f'(3) > 0 \), \( f(x) \) is **increasing** on \( (2, \infty) \).

### Final Answers:

- Critical number \( A = 2 \)
- On \( (-\infty, 2) \): DEC
- On \( (2, \infty) \): INC

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Would you like more details or have any questions?

Here are 5 related questions:
1. How do we determine if a function is concave up or concave down?
2. What are inflection points, and how can we find them?
3. Can we determine whether a critical point is a local maximum or minimum?
4. How do higher-order derivatives affect the shape of the graph?
5. How would the behavior of \( f(x) \) change if the exponent were different?

**Tip:** A function's derivative tells us whether it is increasing or decreasing, so always check the sign of \( f'(x) \) on different intervals to classify the behavior.

Consider the function f(x) = 7 (x - 2)^{2/3}. For this function there are two important intervals: (-∞, A) and (A, ∞) where A is a critical number. Find A. For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC). (-∞, A): (A, ∞):

This detailed solution explains how to find the critical number A for the function f(x) = 7(x - 2)^{2/3}. It covers key calculus concepts such as derivatives, critical points, and determining whether the function is increasing or decreasing in specific intervals. Suitable for students in Grades 11-12.

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