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To find the critical numbers of the function \( f(x) = 2x^{5/3} - 45x^{2/3} \), we need to follow these steps:

### Step 1: Take the derivative of the function
We will use the power rule for derivatives to find \( f'(x) \).

\[
f'(x) = \frac{d}{dx} \left( 2x^{5/3} - 45x^{2/3} \right)
\]

- The derivative of \( 2x^{5/3} \) is:
\[
\frac{d}{dx} \left( 2x^{5/3} \right) = 2 \cdot \frac{5}{3} x^{(5/3) - 1} = \frac{10}{3} x^{2/3}
\]

- The derivative of \( -45x^{2/3} \) is:
\[
\frac{d}{dx} \left( -45x^{2/3} \right) = -45 \cdot \frac{2}{3} x^{(2/3) - 1} = -30x^{-1/3}
\]

Thus, the derivative is:
\[
f'(x) = \frac{10}{3} x^{2/3} - 30x^{-1/3}
\]

### Step 2: Set the derivative equal to zero
To find the critical points, set \( f'(x) = 0 \):

\[
\frac{10}{3} x^{2/3} - 30x^{-1/3} = 0
\]

Multiply through by 3 to eliminate the fractions:
\[
10x^{2/3} - 90x^{-1/3} = 0
\]

### Step 3: Factor the equation
Factor out the common term \( x^{-1/3} \):

\[
x^{-1/3} \left( 10x - 90 \right) = 0
\]

This gives two possibilities:
1. \( x^{-1/3} = 0 \), but this is not possible since \( x^{-1/3} \) is undefined for \( x = 0 \).
2. \( 10x - 90 = 0 \), solve for \( x \):
\[
10x = 90 \quad \Rightarrow \quad x = 9
\]

### Step 4: Analyze undefined points
Since \( f'(x) \) contains the term \( x^{-1/3} \), the derivative is undefined at \( x = 0 \). This is also a critical number.

### Final Answer:
The critical numbers are \( x = 0 \) and \( x = 9 \).

Do you have any questions or need more details? Here are some related questions you might find helpful:

1. How do we identify whether a critical point is a minimum or maximum?
2. What does it mean for a derivative to be undefined at a point?
3. Can a function have critical points where the derivative doesn't equal zero?
4. How do we find inflection points in a function?
5. How do critical points relate to concavity and inflection points?

**Tip:** Always check whether the derivative is undefined at certain points, as they may also be critical points.

Find the critical numbers of the function f(x) = 2x^(5/3) - 45x^(2/3).

Learn how to find the critical numbers for the function f(x) = 2x^(5/3) - 45x^(2/3) using calculus concepts. This step-by-step solution covers finding the derivative, setting it equal to zero, and analyzing points where the derivative is undefined. Ideal for students in Grades 11-12.

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