Question

In the interval $xϵ(e^{2r\pi +\pi /4},e^{2r\pi +5\pi /4})$ in which the function $f\left(x\right)=\sin \left({\log }_{e}x\right)+\cos \left({\log }_{e}x\right)$ is

• A. decreasing
• B. increasing
• C. const.
• D. no monotonicity

Answer 1

The correct option is A decreasing

$f^{\prime }\left(x\right)<0\Rightarrow \frac{\cos \left({\log }_{e}x\right)-\sin \left({\log }_{e}x\right)}{x}<0$ or $\cos t-\sin t<0,t={\log }_{e}x$ or $\sqrt{2}\left[\cos (t+\frac{\pi }{4})\right]<0$
$\therefore t+\frac{\pi }{4}$ lies in 2nd or 3rd quadranti.e., $\frac{\pi }{2}<t+\frac{\pi }{4}<\frac{3\pi }{2}$ or $\frac{\pi }{4}<{\log }_{e}x<\frac{5\pi }{4}$ or $e^{\pi /4}<x<e^{5\pi /4}\therefore xϵ(e^{\pi /4},e^{5\pi /4})$ In general $xϵ(e^{2r\pi +\pi /4},e^{2r\pi +5\pi /4})$