WBJEE · Maths · Application of Derivatives
Let \(g(x)=\int_{x}^{2 x} \frac{f(t)}{t} d t\) where \(x>0\) and \(f\) be continuous function and \(f(2 x)=f(x)\), then
- A \(g(x)\) is strictly increasing function
- B \(\mathrm{g}(\mathrm{x})\) is strictly decreasing function
- C \(\mathrm{g}(\mathrm{x})\) is constant function
- D \(\mathrm{g}(\mathrm{x})\) is not derivable function
Answer & Solution
Correct Answer
(C) \(\mathrm{g}(\mathrm{x})\) is constant function
Step-by-step Solution
Detailed explanation
\(g(x)=\int_{x}^{2 x} \frac{f(t)}{t} d t\) \(g^{\prime}(x)=\frac{f(2 x)}{2 x} \cdot 2^{\prime}-\frac{f(x)}{x} \cdot 1=\frac{f(2 x)-f(x)}{x}=\frac{f(x)-f(x)}{x}[\because f(2 x)=f(x)]\) \(g^{\prime}(x)=0\) \(g(x)=\) constant.
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