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If \(f:(0,\infty)\to\mathbb{R}\) and \(f(x)=x+\frac{1}{x}\), what is the correct conclusion?यदि \(f:(0,\infty)\to\mathbb{R}\) तथा \(f(x)=x+\frac{1}{x}\) हो तो सही निष्कर्ष क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Neither one-one nor ontoन एकैकी न आच्छादी
Step 1: \(f(2)=\frac{5}{2}\) and \(f\left(\frac{1}{2}\right)=\frac{5}{2}\), so it is not one-one. Step 2: Since \(x+\frac{1}{x}\geq2\) for \(x>0\), negative values are not obtained. Step 3: For such functions, compare (x) and \(\frac{1}{x}\).चरण 1: \(f(1)=2\) और \(f\left(\frac{1}{1}\right)=2\) से अलग उदाहरण नहीं मिलता लेकिन \(f(2)=\frac{5}{2}\) और \(f\left(\frac{1}{2}\right)=\frac{5}{2}\) से एकैकीपन टूटता है। चरण 2: \(x+\frac{1}{x}\geq2\) इसलिए ऋणात्मक मान नहीं मिलते। चरण 3: ऐसे फलनों में (x) और \(\frac{1}{x}\) की जोड़ी पर ध्यान दें।
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