TS EAMCET · Maths · Application of Derivatives
Assertion (A) The function \(f(x)=x-\log \left(\frac{1+x}{x}\right), x>0\) has no maximum.Reason (R) If a function \(f(x)\) is strictly increasing in an interval \((a, b)\), then at any point in \((a, b) f^{\prime}(x) \neq 0\) The correct option among the following is
- A (A) is true, (R) is true and (R) is the correct explanation for \(A\).
- B (A) is true, \((R)\) is true but \((R)\) is the not the correct explanation for \(A\).
- C (A) is true but \((R)\) is false.
- D (A) is false but (R) is true.
Answer & Solution
Correct Answer
(A) (A) is true, (R) is true and (R) is the correct explanation for \(A\).
Step-by-step Solution
Detailed explanation
Given function, \(f(x)=x-\log \left(\frac{1+x}{x}\right), x>0\) Differentiating w.r.t. \(x\), we get \(f^{\prime}(x)=\frac{d}{d x}(x)-\frac{d}{d x} \log \left(\frac{1+x}{x}\right)\) \(=1-\frac{1}{\frac{1+x}{x}} \cdot \frac{d}{d x}\left(\frac{1+x}{x}\right) \quad\) [from chain…
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