Question

$y=\sin x-a\sin 2x-\frac{1}{3}\sin 3x+2ax,$ then y increases for all values of x if $a>k.$, what is k?

Answer 1

Yes. $\frac{dy}{dx}=\cos x-2a\cos 2x-\cos 3x+2a$
$=2\sin 2x\sin x+2a(1-\cos 2x)$
$=4{\sin }^{2}x\cos x+4a{\sin }^{2}x$
$=4{\sin }^{2}x(\cos x+a)$ Now $4{\sin }^{2}x$ is +ive and if $\cos x+a>0$ then f(x) increases for all value of x.Now the least value of $\cos x$ is -1.$\therefore -1+a>0\Rightarrow a>1.$