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Given the function \( f(x) = (x - 2)^{1/3} \), let's analyze and find the answers for each part:

### (a) Axis of Symmetry, Vertical/Horizontal Asymptotes
- **Axis of Symmetry**: This function does not have an axis of symmetry since it’s an odd root function, which means it doesn’t reflect symmetrically around any line.
- **Vertical Asymptote**: There is no vertical asymptote. The function \( f(x) = (x - 2)^{1/3} \) is defined for all real \( x \) values.
- **Horizontal Asymptote**: There is no horizontal asymptote either because as \( x \) approaches infinity or negative infinity, \( f(x) \) also goes to infinity or negative infinity, respectively.

### (b) Intervals of Increasing/Decreasing
To find intervals where the curve is increasing or decreasing, we need the first derivative:
1. Differentiate \( f(x) = (x - 2)^{1/3} \):
   \[
   f'(x) = \frac{1}{3}(x - 2)^{-2/3} = \frac{1}{3(x - 2)^{2/3}}
   \]
2. **Sign of \( f'(x) \)**:
   - For \( x > 2 \), \( f'(x) > 0 \), indicating the function is increasing.
   - For \( x < 2 \), \( f'(x) > 0 \) as well, so the function is also increasing for values less than 2.
   - At \( x = 2 \), \( f'(x) \) is undefined, but it doesn’t change the nature of the function since the values are still increasing on either side.

   **Conclusion**: The function is increasing on \( (-\infty, \infty) \).

### (c) Maximum/Minimum Values or Inflection Points
- Since \( f(x) \) is always increasing, there are no local maximum or minimum points.
- To check for inflection points, we need the second derivative:
  \[
  f''(x) = -\frac{2}{9}(x - 2)^{-5/3} = -\frac{2}{9(x - 2)^{5/3}}
  \]
- **Concavity**:
  - For \( x > 2 \), \( f''(x) < 0 \), indicating concave down.
  - For \( x < 2 \), \( f''(x) > 0 \), indicating concave up.
  - At \( x = 2 \), \( f''(x) \) is undefined, so \( x = 2 \) is an inflection point.

### (d) Intervals of Concavity
- **Concave Up**: \( (-\infty, 2) \)
- **Concave Down**: \( (2, \infty) \)

### (e) Sketch of \( f(x) \)
To sketch \( f(x) = (x - 2)^{1/3} \):
- The graph passes through \( (2, 0) \)

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).

Explore a detailed analysis of the function f(x) = (x - 2)^(1/3), including symmetry, vertical and horizontal asymptotes, intervals of increase/decrease, maximum/minimum values, concavity, and inflection points. Ideal for students studying calculus and advanced function analysis in Grades 11-12.

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