Mathematics

This problem hinges on recognizing that the inequality is actually tight—it becomes an equality at the limit point. The key insight is to treat $f(x)$ as an unknown that satisfies a constraint, then use the AM-GM inequality backwards: the sum $f(x) + \frac{e^{x+x^2}}{f(x)}$ is minimized when these two terms are equal. Since the right side is fixed, equality must hold at $x=1$, which forces $f(1) = \sqrt{e^{1+1}} = e$.

This problem hinges on recognizing that the inequality is actually tight—it becomes an equality at the limit point. The key insight is to treat $f(x)$ as an unknown that satisfies a constraint, then use the AM-GM inequality backwards: the sum $f(x) + \frac{e^{x+x^2}}{f(x)}$ is minimized when these two terms are equal. Since the right side is fixed, equality must hold at $x=1$, which forces $f(1) = \sqrt{e^{1+1}} = e$.