MHT CET · Maths · Application of Derivatives
\(\mathrm{F}(\mathrm{x})=\log |\sin \mathrm{x}|\), where \(\mathrm{x} \in(0, \pi)\) is strictly increasing on
- A \(\left(\frac{\pi}{2}, \pi\right)\) only
- B \((0, \pi)\) only
- C \(\left(0, \frac{\pi}{2}\right)\) only
- D \(\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)\) only
Answer & Solution
Correct Answer
(C) \(\left(0, \frac{\pi}{2}\right)\) only
Step-by-step Solution
Detailed explanation
\(f(x)=\log |\sin x|\), where \(x \in(0, \pi)\)
\(\therefore \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\sin \mathrm{x}} \times \cos \mathrm{x}=\cot \mathrm{x}\)
When \(\mathrm{f}^{\prime}(\mathrm{x})>0\), we say \(\frac{\cos \mathrm{x}}{\sin \mathrm{x}}>0\)
Here \(\sin x>0 \ldots[x \in(0, \pi)]\)
for the function to be strictly increasing, \(\cos x>0\)
\(\rightarrow \mathrm{x} \in\left(0, \frac{\pi}{2}\right)\) only.
\(\therefore \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\sin \mathrm{x}} \times \cos \mathrm{x}=\cot \mathrm{x}\)
When \(\mathrm{f}^{\prime}(\mathrm{x})>0\), we say \(\frac{\cos \mathrm{x}}{\sin \mathrm{x}}>0\)
Here \(\sin x>0 \ldots[x \in(0, \pi)]\)
for the function to be strictly increasing, \(\cos x>0\)
\(\rightarrow \mathrm{x} \in\left(0, \frac{\pi}{2}\right)\) only.
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