Let the function $f : \mathbb{R} \to \mathbb{R}$ be defined by
$$f(x) = \dfrac{\sin x}{e^{\pi x}} \dfrac{x^{2023} + 2024x + 2025}{x^2 - x + 3} + \dfrac{2}{e^{\pi x}} \dfrac{x^{2023} + 2024x + 2025}{x^2 - x + 3}.$$\nThen the number of solutions of $f(x) = 0$ in $\mathbb{R}$ is ___.
Step-by-Step Solution
Key Concept: Analyzing zeros of functions by factoring out strictly positive terms and checking monotonicity of the remaining polynomial.
Simplify the function expression:
$$f(x) = \dfrac{(\sin x + 2)(x^{2023} + 2024x + 2025)}{e^{\pi x}(x^2 - x + 3)}$$
For $f(x) = 0$, we analyze each component:
1) $\sin x + 2 \ge 1 > 0$ for all $x \in \mathbb{R}$, so this term is never 0.
2) $e^{\pi x} > 0$ for all $x \in \mathbb{R}$.
3) $x^2 - x + 3$ has discriminant $\Delta = 1 - 12 = -11 < 0$, so it is strictly positive for all $x \in \mathbb{R}$.
Thus, $f(x) = 0$ if and only if:
$$g(x) = x^{2023} + 2024x + 2025 = 0$$
To find the number of real roots of $g(x)$, we compute its derivative:
$$g'(x) = 2023x^{2022} + 2024$$
Since $x^{2022} \ge 0$, $g'(x) \ge 2024 > 0$ for all $x \in \mathbb{R}$. Thus, $g(x)$ is a strictly increasing function on $\mathbb{R}$.
Since $g(x)$ is continuous and strictly increasing, it must have exactly one real root.
Thus, the number of solutions is 1.
Correct Answer: 1