Let f : (0, \infty) → R be a differentiable function such that f '(x) = 2 – \frac{f \left(\right. x \left.\right)}{x} for all x \in (0, \infty) and f (1) \neq 1. Then

Engineering

Mathematics

Linear Differential Equation

Question

Let f : (0, $\infty$ ) → 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

$\underset{x \to 0^{+}}{Lim} xf (\frac{1}{x}) = 2$

$\underset{x \to 0^{+}}{Lim} x^{2} f' (x) = 0$

| f (x) | $\leq$ 2 for all x $\in$ (0, 2)

$\underset{x \to 0^{+}}{Lim} f' (\frac{1}{x}) = 1$

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$f' (x) = 2 - \frac{f (x)}{x}$

x f ' (x) + f (x) = 2x

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

x f (x) = x² + C

$f (x) = x + \frac{C}{x}$ $f (1) \neq 1$ $C \neq 0$

(A) $\underset{x \to 0^{+}}{Lim} f' (\frac{1}{x}) = \underset{x \to 0^{+}}{Lim} (1 - {Cx}^{2}) = 1 \Rightarrow (A) is correct$

(B) $\underset{x \to 0^{+}}{Lim} x f (\frac{1}{x}) = \underset{x \to 0^{+}}{Lim} (1 + {Cx}^{2}) = 1 \Rightarrow (B) is incorrect$

(C) $\underset{x \to 0^{+}}{Lim} x^{2} f' (x) = \underset{x \to 0^{+}}{Lim} x^{2} (1 - \frac{C}{x^{2}}) = - C \neq 0 \Rightarrow (C) is incorrect$

(D) $| f (x) | = | x + \frac{C}{x} | = | x | + \frac{| C |}{| x |} \leq 2 \sqrt{| C |} depends on C .$