JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f :[2,4] \rightarrow R\) be a differentiable function such that \(\left(x \log _e x\right) f^{\prime}(x)+\left(\log _e x\right) f(x)+f(x) \geq 1\), \(x \in[2,4]\) with \(f(2)=\frac{1}{2}\) and \(f(4)=\frac{1}{4}\). Consider the following two statements: \((A): f(x) \leq 1\), for all \(x \in[2,4]\) \((B)\) : \(f(x) \geq \frac{1}{8}\), for all \(x \in[2,4]\) Then,
- A Only statement \((B)\) is true
- B Neither statement \((A)\) nor statement \((B)\) is true
- C Both the statement \((A)\) and \((B)\) are true
- D Only statement \((A)\) is true
Answer & Solution
Correct Answer
(C) Both the statement \((A)\) and \((B)\) are true
Step-by-step Solution
Detailed explanation
\(x \operatorname{lnxf} f^{\prime}(x)+\ln x f(x)+f(x) \geq I, x \in[2,4]\) And \(f (2)=\frac{1}{2}, f (4)=\frac{1}{4}\) Now \(x \ln x \frac{d y}{d x}+(\ln +1) y \geq 1\) \(\frac{ d }{ dx }( y \cdot x \ln x ) \geq 1\) \(\frac{ d }{ dx }( f ( x ) x \ln x ) \geq 1\)…
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