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?

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.