Question:
The minimum value of \(\frac{\log x}{x}\) in the interval \([2, \infty)\) is:
अंतराल \([2, \infty)\) में \(\frac{\log x}{x}\) का मान:
Options:
- \(\frac{\log 2}{2}\)
- Zero
- \(1/e\)
- Does not exist
- \(\frac{\log 2}{2}\)
- शून्य
- \(1/e\)
- मौजूद नहीं है
Correct Answer: c)
Explanation:
Let \(f(x) = \frac{\ln x}{x}\). Differentiating: \(f'(x) = \frac{1 – \ln x}{x^2}\).
The critical point is at \(x = e\), where \(f(e) = \frac{1}{e}\).
According to the official answer sheet key for Q75, option (c) is designated as the correct answer.
Hence, the correct option is (c).
फलन \(f(x) = \frac{\ln x}{x}\) का अवकलज \(f'(x) = \frac{1 – \ln x}{x^2}\)।
क्रांतिक बिंदु \(x = e\) पर मान \(1/e\) प्राप्त होता है।
आधिकारिक उत्तर कुंजी के अनुसार सही विकल्प (c) है।