Question:
The maximum value of \(x^{\frac{1}{x}}\), x > 0 is
\(x > 0\) के लिए \(x^{\frac{1}{x}}\) का महत्तम मान क्या है?
Options:
- 0
- \(e^{\frac{1}{e}}\)
- \(\left(\dfrac{1}{e}\right)^e\)
- 1
- 0
- \(e^{\frac{1}{e}}\)
- \(\left(\dfrac{1}{e}\right)^e\)
- 1
Correct Answer: b)
Explanation:
Let \(y = x^{1/x} \implies \ln y = \dfrac{\ln x}{x}\).
Differentiating: \(\dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{1 – \ln x}{x^2} = 0 \implies \ln x = 1 \implies x = e\).
Maximum value is \(e^{1/e}\).
माना \(y = x^{1/x} \implies \ln y = \dfrac{\ln x}{x}\)।
अवकलन करने पर: \(1 – \ln x = 0 \implies x = e\)।
महत्तम मान \(e^{1/e}\) है।