Question

If the maximum and the minimum values of $1+\sin (\frac{\pi }{4}+\theta )+2\cos (\frac{\pi }{4}-\theta )$ for all real values of $\theta$ are $\lambda$ and $\mu$ respectively, then $\lambda -\mu$ is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is C $6$

$\sin (\frac{\pi }{4}+\theta )=\cos (\frac{\pi }{2}-(\frac{\pi }{4}+\theta ))=\cos (\frac{\pi }{4}-\theta )$
$\therefore 1+\sin (\frac{\pi }{4}+\theta )+2\cos (\frac{\pi }{4}-\theta )$
$=1+3\cos (\frac{\pi }{4}-\theta )$

$-3\le 3\cos (\frac{\pi }{4}-\theta )\le 3$
$\Rightarrow -2\le 1+3\cos (\frac{\pi }{4}-\theta )\le 4$
$\therefore \lambda =4$ and $\mu =-2$
$\Rightarrow \lambda -\mu =6$