Let f: R − {0} → R be a function satisfying f(x/y) = f(x)/f(y) for all x, y, f(y) ≠ 0.

Question

Let \(\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) be a function satisfying \(f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}\) for all \(x, y, f(y) \neq 0\). If \(f^{\prime}(1)=2024\), then

(1) \({xf}^{\prime}({x})-2024 {f}({x})=0\)

(2) \(x f^{\prime}(x)+2024 f(x)=0\)

(3) \({xf}^{\prime}({x})+{f}({x})=2024\)

(4) \(x f^{\prime}(x)-2023 f(x)=0\)

Answer 1

Correct option is (1) \({xf}^{\prime}({x})-2024 {f}({x})=0\)

\(f\left( \frac xy\right) = \frac{f(x)}{ f(y)}\)

\(f'(1) = 2024\)

\(f(1) = 1\)

Partially differentiating w. r. t. x

\(f^{\prime}\left(\frac{x}{y}\right) \cdot \frac{1}{y}=\frac{1}{f(y)} f^{\prime}(x)\)

\({y} \rightarrow {x}\)

\({f}^{\prime}(1) \cdot \frac{1}{{x}}=\frac{{f}^{\prime}({x})}{{f}({x})} \)

\(2024 f(x)=x f^{\prime}(x) \Rightarrow f^{\prime}(x)-2024 f(x)=0\)