AP EAMCET · Maths · Inverse Trigonometric Functions
The interval in which the function \(f(x)=\operatorname{Tan}^{-1}(\sin x+\cos x)\) is an increasing function, is
- A \(\left(0, \frac{\pi}{2}\right)\)
- B \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
- C \(\left(-\frac{3 \pi}{4}, \frac{\pi}{4}\right)\)
- D \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\)
Answer & Solution
Correct Answer
(C) \(\left(-\frac{3 \pi}{4}, \frac{\pi}{4}\right)\)
Step-by-step Solution
Detailed explanation
\(f'(x) = \frac{1}{1+(\sin x+\cos x)^2} (\cos x - \sin x)\) \(f'(x) > 0 \implies \cos x - \sin x > 0\) \(\cos x > \sin x\) \(\cos x - \sin x = \sqrt{2}\cos(x+\frac{\pi}{4})\) \(\sqrt{2}\cos(x+\frac{\pi}{4}) > 0 \implies \cos(x+\frac{\pi}{4}) > 0\)…
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