1. **Problem Statement:** We want to analyze the function $$f(x) = x^2 + \frac{1024}{x}$$ and understand its behavior graphically. 2. **Formula and Rules:** The function is a combination of a quadratic term $$x^2$$ and a rational term $$\frac{1024}{x}$$. Note that $$x \neq 0$$ because division by zero is undefined. 3. **Intermediate Work:** - The function can be written as $$f(x) = x^2 + 1024x^{-1}$$. - To find critical points, we differentiate: $$f'(x) = 2x - \frac{1024}{x^2}$$. - Set derivative to zero to find extrema: $$2x - \frac{1024}{x^2} = 0 \implies 2x = \frac{1024}{x^2} \implies 2x^3 = 1024 \implies x^3 = 512 \implies x = 8$$. 4. **Explanation:** - The function has a critical point at $$x=8$$. - For $$x > 0$$, the function is defined and smooth. - For $$x < 0$$, the function is also defined but behaves differently due to the negative values. - The function is undefined at $$x=0$$, so there is a vertical asymptote there. 5. **Summary:** The function $$f(x) = x^2 + \frac{1024}{x}$$ has a vertical asymptote at $$x=0$$ and a critical point at $$x=8$$ where the function has an extremum. Final answer: The function is $$f(x) = x^2 + \frac{1024}{x}$$ with critical point at $$x=8$$ and undefined at $$x=0$$.