Question

For the function $f\left(x\right)=\sin x+2\cos x,$ which of the following is true regarding its monotonicity at $x=\frac{\pi }{4}$

• A. Increasing
• B. Decreasing
• C. Non-monotonic
• D. Can't be determined

Answer 1

The correct option is B Decreasing

$f^{\prime }\left(x\right)=\cos x-2\sin x$
$f^{\prime }\left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}-2\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{\sqrt{2}}$

Clearly $f^{\prime }\left(\frac{\pi }{4}\right)<0\Rightarrow f\left(x\right)$ is decreasing at $x=\frac{\pi }{4}$