JEE Mains · Maths · STD 12 - 6. Application of derivatives
The interval in which the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}}, \mathrm{x}>0\), is strictly increasing is
- A \(\left(0, \frac{1}{\mathrm{e}}\right]\)
- B \(\left[\frac{1}{\mathrm{e}^2}, 1\right)\)
- C \((0, \infty)\)
- D \(\left[\frac{1}{\mathrm{e}}, \infty\right)\)
Answer & Solution
Correct Answer
(D) \(\left[\frac{1}{\mathrm{e}}, \infty\right)\)
Step-by-step Solution
Detailed explanation
\( f(x)=x^x ; x>0 \) \( \ell n y=x \ell n x \) \( \frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\ell n x \) \( \frac{d y}{d x}=x^x(1+\ell n x) \) for strictly increasing \( \frac{d y}{d x} \geq 0 \Rightarrow x^x(1+\ell n x) \geq 0 \) \( \Rightarrow \ell n x \geq-1 \)…
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