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The problem you're working on involves analyzing a function \( f(x) = x^{3/5}(x - 7) \), and you're asked to determine critical points, intervals of increasing/decreasing behavior, concavity, and more. Let's walk through the steps to solve it.

### 1. **Critical Points** (A and B)
The critical points of a function occur where the derivative is zero or undefined. The given function is:
\[ f(x) = x^{3/5}(x - 7) \]

#### Step 1: Differentiate \( f(x) \)
Using the product rule for differentiation:

\[
f'(x) = \frac{3}{5}x^{-2/5}(x - 7) + x^{3/5}(1)
\]

Simplify this:

\[
f'(x) = \frac{3}{5}x^{-2/5}(x - 7) + x^{3/5}
\]

To make the next steps easier, let's factor the expression. Factoring out \( x^{-2/5} \):

\[
f'(x) = x^{-2/5} \left( \frac{3}{5}(x - 7) + x \right)
\]

Simplifying inside the parentheses:

\[
f'(x) = x^{-2/5} \left( \frac{3}{5}x - \frac{21}{5} + x \right)
= x^{-2/5} \left( \frac{8}{5}x - \frac{21}{5} \right)
\]

Finally, rewrite the derivative as:

\[
f'(x) = \frac{1}{5}x^{-2/5}(8x - 21)
\]

#### Step 2: Set \( f'(x) = 0 \) to find critical points
For the product to be zero, either:

1. \( 8x - 21 = 0 \Rightarrow x = \frac{21}{8} \) (Critical point B)
2. \( x^{-2/5} = 0 \) (undefined at \( x = 0 \), so critical point A is at \( x = 0 \))

Thus, the critical points are:
- \( A = 0 \)
- \( B = \frac{21}{8} \)

### 2. **Intervals of Increasing and Decreasing**
To determine where the function is increasing or decreasing, evaluate the sign of \( f'(x) \) on intervals around the critical points.

The critical points divide the real line into intervals:
- \( (-\infty, 0) \)
- \( (0, \frac{21}{8}) \)
- \( (\frac{21}{8}, \infty) \)

Test a point in each interval to determine the sign of \( f'(x) \) in that region.

- For \( x < 0 \), \( f'(x) \) is negative, so the function is decreasing on \( (-\infty, 0) \).
- For \( 0 < x < \frac{21}{8} \), \( f'(x) \) is positive, so the function is increasing on \( (0, \frac{21}{8}) \).
- For \( x > \frac{21}{8} \), \( f'(x) \) is positive, so the function is increasing on \( (\frac{21}{8}, \infty) \).

### 3. **Concavity and Inflection Points** (C and D)
To determine concavity, we need the second derivative \( f''(x) \).

#### Step 1: Differentiate \( f'(x) \)
Using the product rule again on the first derivative:

\[
f'(x) = \frac{1}{5}x^{-2/5}(8x - 21)
\]

Differentiate to get \( f''(x) \). After doing so, find the points where \( f''(x) = 0 \) or is undefined to get the inflection points, denoted by C and D.

Would you like me to continue with the second derivative and concavity analysis, or do you have any questions so far?

### Questions for Further Exploration:
1. How do critical points relate to local extrema in a function?
2. Can you explain the meaning of concavity and how inflection points are identified?
3. How does the behavior of the function near critical points help in sketching the graph of \( f(x) \)?
4. What is the importance of the second derivative in determining concavity?
5. Can you explain how to apply the first derivative test to classify critical points?

### Tip:
When working on increasing/decreasing intervals, always use test points in the derivative expression to confirm the sign changes around the critical points.

Consider the function f(x) = x^(3/5)(x - 7). This function has two critical numbers A < B. Determine the critical points, increasing/decreasing intervals, concavity, and inflection points.

Learn how to analyze the function f(x) = x^(3/5)(x - 7), find its critical points, intervals of increase/decrease, concavity, and inflection points. This problem applies first and second derivative tests. Suitable for undergraduate students in Calculus I.

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