1. **Stating the problem:** Determine on which interval the function $f(x) = x^2$ is convex downward (concave up). 2. **Recall the definition:** A function is concave up on an interval if its second derivative is positive on that interval. 3. **Find the first derivative:** $$f'(x) = 2x$$ 4. **Find the second derivative:** $$f''(x) = 2$$ 5. **Analyze the second derivative:** Since $f''(x) = 2$ is positive for all real numbers $x$, the function $f(x) = x^2$ is concave up on the entire real line. 6. **Conclusion:** The function is concave up on $\mathbb{R}$. **Answer:** D) $\mathbb{R}$