AP EAMCET · Maths · Application of Derivatives
The interval in which the curve represented by \(f(x)=2 x+\log \left(\frac{x}{2+x}\right)\) is increasing is
- A \((-\infty, 0)\)
- B \((-2, \infty)\)
- C \((-\infty,-2) \cup(0, \infty)\)
- D \((-2,0)\)
Answer & Solution
Correct Answer
(C) \((-\infty,-2) \cup(0, \infty)\)
Step-by-step Solution
Detailed explanation
\(f'(x) = \frac{d}{dx}\left(2x + \log x - \log(2+x)\right)\) \(= 2 + \frac{1}{x} - \frac{1}{2+x}\) \(= \frac{2x(2+x) + (2+x) - x}{x(2+x)}\) \(= \frac{2x^2 + 4x + 2}{x(2+x)}\) \(= \frac{2(x+1)^2}{x(2+x)}\) For \(f(x)\) to be increasing, \(f'(x) > 0\).…
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