Let $f:(0,\infty)\to\mathbb{R}$ be a differentiable function such that $f'(x)=2-\frac{f(x)}{x}$ for all $x\in(0,\infty)$ and $f(1)\neq 1$. Then

Let f:(0,∞)→R be a differentiable function such that f'(x)=2-f(x)/x for all x∈(0,∞) and f(1)≠ 1. Then

$f'(x)=2-\dfrac{f(x)}{x}, \quad x\in(0,\infty), \quad f(1)\neq 1$

$\text{To find: which statement(s) is/are correct}$

$f'(x)+\dfrac{1}{x}f(x)=2, \quad \text{IF}=x$

$\dfrac{d}{dx}\big(x f(x)\big) = 2x$

$x f(x) = x^2 + C \;\Rightarrow\; f(x) = x + \dfrac{C}{x}, \quad C\neq 0$

$f'(x) = 1-\dfrac{C}{x^2} \;\Rightarrow\; f'\!\left(\dfrac{1}{x}\right) = 1-Cx^2 \to 1$

$\Rightarrow (1) \text{ is correct}$

$x\,f\!\left(\dfrac{1}{x}\right) = x\left(\dfrac{1}{x}+Cx\right) = 1+Cx^2 \to 1 \neq 2$

$\Rightarrow (2) \text{ is incorrect}$

$x^2 f'(x) = x^2\left(1-\dfrac{C}{x^2}\right) = x^2 - C \to -C \neq 0$

$\Rightarrow (3) \text{ is incorrect}$

$f(x)=x+\dfrac{C}{x}\to\pm\infty \text{ as } x\to 0^{+} \;\Rightarrow\; |f|\le 2 \text{ fails}$

$\Rightarrow (4) \text{ is incorrect}$

$\boxed{\text{Only statement (1) is correct}}$

$\text{Answer: (1)}$