Question

The function $y=\frac{4\cos \theta }{\sin \theta +2}\forall \theta \in [0,\frac{\pi }{2}]$ is

• A. always increasing
• B. always decreasing
• C. increasing in $[0,\frac{\pi }{4}]$
• D. decreasing in $[0,\frac{\pi }{4}]$

Answer 1

The correct option is B always decreasing

$y=\frac{4\cos \theta }{\sin \theta +2}\Rightarrow y^{\prime }=\frac{-4\sin \theta (\sin \theta +2)-4\cos \theta \left(\cos \theta \right)}{(\sin \theta +2{)}^2}\Rightarrow y^{\prime }=\frac{-4-8\sin \theta }{(\sin \theta +2{)}^2}$

For $\theta \in [0,\frac{\pi }{2}];\sin \theta \ge 0$
So $y^{\prime }<0\forall \theta \in [0,\frac{\pi }{2}]$
Therefore the function is always decreasing.