Question

For the function $f\left(x\right)=sin x+2cos⁡x, \, \forall x\in \left[0,2 \pi \right]$ , we obtain

• A. a local point of maxima at $x=\alpha $ , where $\alpha $ is in $1^{s t}$ quadrant
• B. a local point of maxima at $x=\alpha $ , where $\alpha $ is in $3^{r d}$ quadrant
• C. a local point of minima at $x=\alpha $ , where $\alpha $ is in $1^{s t}$ quadrant
• D. a local point of minima at $x=\alpha $ , where $\alpha $ is in $2^{n d}$ quadrant

Answer 1

Answer: (A)

$f\left(x\right)=sin x+2cos⁡x,\forall x\in \left[0,2 \pi \right]$
$f^{'}\left(x\right)=cos x-2sin⁡x$
$\Rightarrow f^{'}\left(x\right)$ changes its sign from positive to negative at $x=tan^{- 1}\left(\frac{1}{2}\right)\in \left(0 , \frac{\pi }{2}\right)$
So at this point, local maxima occurs
$\Rightarrow f^{'}\left(x\right)$ changes its sign from negative to positive at $x=\pi +tan^{- 1}\left(\frac{1}{2}\right)\in \left(\pi , \frac{3 \pi }{2}\right)$
So at this point, local minima occurs