1. **Stating the problem:** We want to analyze and graph the function $$f(x) = \frac{x^2}{(x-1)^2}$$ with the given properties about domain, monotonicity, local minima, concavity, inflection points, and asymptotes. 2. **Domain:** The function is undefined at $$x=1$$ because the denominator is zero there. So the domain is $$(-\infty, 1) \cup (1, \infty)$$. 3. **Vertical asymptote:** Since the denominator is zero at $$x=1$$ and the numerator is not zero there, there is a vertical asymptote at $$x=1$$. 4. **Horizontal asymptote:** For large $$|x|$$, $$f(x) \approx \frac{x^2}{x^2} = 1$$, so the horizontal asymptote is $$y=1$$. 5. **Monotonicity:** Given that $$f$$ is increasing on $$(0,1)$$ and decreasing on $$(-\infty,0) \cup (1, \infty)$$, the function has a local minimum at $$x=0$$ with $$f(0) = 0$$. 6. **Concavity and inflection point:** The function is concave down on $$(-\infty, -\frac{1}{2})$$ and concave up on $$(-\frac{1}{2}, 1) \cup (1, \infty)$$. The inflection point is at $$\left(-\frac{1}{2}, \frac{1}{9}\right)$$. 7. **Summary:** - Domain: $$(-\infty, 1) \cup (1, \infty)$$ - Vertical asymptote: $$x=1$$ - Horizontal asymptote: $$y=1$$ - Increasing on $$(0,1)$$ - Decreasing on $$(-\infty,0) \cup (1, \infty)$$ - Local minimum at $$(0,0)$$ - Concave down on $$(-\infty, -\frac{1}{2})$$ - Concave up on $$(-\frac{1}{2}, 1) \cup (1, \infty)$$ - Inflection point at $$\left(-\frac{1}{2}, \frac{1}{9}\right)$$ This matches the given conditions and fully describes the graph of $$f(x)$$.