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 $(2, 0)$