Question

Find the values of ${}^{\prime }{a}^{\prime }$ for which the function $f\left(x\right)=\sin x-a\sin 2x-\frac{1}{3}\sin 3x+2ax$ increases throughout the number line.

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{5}{2}$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$

$f\left(x\right)=\sin x-a\sin 2x-\frac{1}{3}\sin 3x+2ax$

$f\left(x\right)=\cos x-2a\cos 2x-\cos 3x+2a$

$=\cos x-cos3x+2a(1-\cos 2x)$

for $f\left(x\right)$ to be increasing

$f^{\prime }\left(x\right)$ should be positive.

$[-2,2]$ $[0,2]$

$\cos x-cos3x+2a(1-\cos 2x)>0$

$\Rightarrow 2a\times 2=2$

$a=\frac{1}{2}$