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We have a function \( f(x) = (x - 2)^{\frac{1}{3}} \) with several questions about its properties:

1. **Axis of symmetry, vertical asymptote, horizontal asymptote (if any)**  
   Since this function is defined by a cube root, it has no vertical or horizontal asymptotes. It’s also not symmetric about any particular axis, as it’s neither an even nor odd function.

2. **Intervals where the curve is increasing/decreasing**  
   To determine where the function is increasing or decreasing, we find \( f'(x) \). For \( f(x) = (x - 2)^{\frac{1}{3}} \), we use the power rule to get:
   \[
   f'(x) = \frac{1}{3} (x - 2)^{-\frac{2}{3}}
   \]
   The derivative \( f'(x) > 0 \) for all \( x \neq 2 \), indicating the function is increasing for all \( x \neq 2 \).

3. **Maximum/minimum values or inflection points**  
   Since \( f'(x) \) does not change sign, there are no local maximum or minimum points. Additionally, at \( x = 2 \), \( f(x) \) has a cusp rather than a smooth curve, so there’s no inflection point there.

4. **Intervals where the curve is concave down/up**  
   To analyze concavity, we find the second derivative \( f''(x) \):
   \[
   f''(x) = -\frac{2}{9} (x - 2)^{-\frac{5}{3}}
   \]
   For \( x < 2 \), \( f''(x) > 0 \), so the curve is concave up, and for \( x > 2 \), \( f''(x) < 0 \), indicating concave down.

5. **Sketch of \( f(x) \)**  
   The sketch would show a cusp at \( x = 2 \), with the curve increasing to the left and right of this point. It’s concave up for \( x < 2 \) and concave down for \( x > 2 \).

Would you like detailed calculations or further clarification on any of these points?

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### Related Questions
1. How does the cube root affect the continuity of the function?
2. Can we determine the end behavior of \( f(x) \) as \( x \to \pm \infty \)?
3. How would the sketch change if the function were \( (x - 2)^{1/2} \) instead?
4. How does concavity affect the shape of a graph around a critical point?
5. Can a function have both an asymptote and a point of inflection?

### Tip
When analyzing functions like \( (x - a)^{1/n} \), pay attention to behavior around \( x = a \) and whether \( n \) is even or odd—it significantly affects symmetry and concavity.

Diberikan fungsi f(x) = (x - 2)^(1/3). Dapatkan:
(a) sumbu simetri, asimtot tegak/asimtot datar, jika ada
(b) interval dimana kurva naik/turun
(c) nilai maksimum/minimum atau titik belok
(d) interval dimana kurva cekung ke bawah/ke atas
(e) sketsa grafik f(x).

Analyze the function f(x) = (x - 2)^(1/3), covering its axis of symmetry, vertical and horizontal asymptotes, intervals of increase/decrease, maximum/minimum points, concavity, and sketching. Suitable for advanced high school mathematics.

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