JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(\mathrm{f}:(-1, \infty) \rightarrow \mathrm{R}\) be defined by \(\mathrm{f}(0)=1\) and \(f(x)=\frac{1}{x} \log _{e}(1+x), x \neq 0 .\) Then the function \(f\)
- A decreases in \((-1, \infty)\)
- B decreases in \((-1,0)\) and increases in \((0, \infty)\)
- C increases in \((-1, \infty)\)
- D increases in \((-1,0)\) and decreases in \((0, \infty)\)
Answer & Solution
Correct Answer
(A) decreases in \((-1, \infty)\)
Step-by-step Solution
Detailed explanation
\(f^{\prime}(x)=\frac{\frac{x}{1+x}-\ell n(1+x)}{x^{2}}\) \(=\frac{x-(1+x) \ell n(1+x)}{x^{2}(1+x)}\) Suppose \(\mathrm{h}(\mathrm{x})=\mathrm{x}-(1+\mathrm{x}) \ell \mathrm{n}(1+\mathrm{x})\)…
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