Question

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

• A. a local point of maxima at $x=\alpha$ , where $\alpha$ is in $1^{st}$ quadrant
• B. a local point of maxima at $x=\alpha$ , where $\alpha$ is in $3^{rd}$ quadrant
• C. a local point of minima at $x=\alpha$ , where $\alpha$ is in $1^{st}$ quadrant
• D. a local point of minima at $x=\alpha$ , where $\alpha$ is in $2^{nd}$ quadrant

Answer 1

Answer: (A)

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