Question

The function $f\left(x\right)=ta{n}^{-1}(sinx+cosx),x>0$ is always an increasing function on the interval

• A. $(0,\pi )$
• B. $(0,\frac{\pi }{2})$
• C. $(0,\frac{\pi }{4})$
• D. $(0,\frac{3\pi }{4})$
• E. $(0,\frac{5\pi }{4})$

Answer 1

The correct option is C $(0,\frac{\pi }{4})$

$f\left(x\right)=y=ta{n}^{-1}\left(\sqrt{2}sin(x+\frac{\pi }{4})\right)\Rightarrow tany=\sqrt{2}sin(x+\frac{\pi }{4})\Rightarrow se{c}^{2}y\frac{dy}{dx}=\sqrt{2}cos(x+\frac{\pi }{4})\frac{dy}{dx}>0\Rightarrow cos(x+\frac{\pi }{4})>0.\therefore x\in (0,\frac{\pi }{4})$