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If (f:\mathbb{R}\to\mathbb{R}), (f(x)=\frac{x}{x^2+1}), what is the correct reason that (f) is not one-one?यदि (f:\mathbb{R}\to\mathbb{R}), (f(x)=\frac{x}{x^2+1}) है, तो (f) के एकैकी न होने का सही कारण कौन-सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(f(2)=f(\frac{1}{2})\) / (f(2)=f(\frac{1}{2}))\(f(2)=f(\frac{1}{2})\) / (f(2)=f(\frac{1}{2}))
Step 1: To prove not one-one, it is enough to find two different inputs with the same output. Step 2: (f(2)=\frac{2}{5}) and (f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{4}+1}=\frac{2}{5}). Step 3: In such questions, finding a smart pair can be faster than using derivatives.चरण 1: एकैकी न होने के लिए दो अलग इनपुटों का समान मान दिखाना काफी है। चरण 2: (f(2)=\frac{2}{5}) और (f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{4}+1}=\frac{2}{5})। चरण 3: ऐसे प्रश्नों में उपयुक्त जोड़ी खोजना अवकलज से भी तेज हो सकता है।
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