Let the function f : R → R be defined by f(x) = sinx/e^πx (x^2023 + 2024x + 2025) / (x^2 − x + 3) + 2/e^πx (x^2023 + 2024x + 2025) / (x^2 − x + 3)

Question

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by

\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^{2}-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^{2}-x+3\right)}\)

Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is ______.

Answer 1

Correct answer: 1

\(f(x)=0\)

\(\Rightarrow \frac{x^{2023}+2024 x+2025}{\left(x^{2}-x+3\right)}\left[\frac{\sin x+2}{e^{\pi x}}\right]=0\)

\(\Rightarrow x^{2023}+2024 x+2025=0\)

Let \(g(x)=x^{2023}+2024 x+2025\)

\(g^{\prime}(x)=2023 x^{2022}+2024>0 \ \forall x \in \mathbb{R}\)

\(\therefore f(x)=0\) has only one solution.