Đặt: $f(x)=3\sin x+4\cos x+mx-2$.
Ta có: $f\left(\dfrac{\pi}{3}\right)=\dfrac{m\pi}{3}+\dfrac{3\sqrt3}{2}; f\left(\dfrac{-\pi}{3}\right)=\dfrac{-m\pi}{3}-\dfrac{3\sqrt3}{2}$
SUy ra: $f\left(\dfrac{\pi}{3}\right)f\left(\dfrac{-\pi}{3}\right)=-\left(\dfrac{m\pi}{3}+\dfrac{3\sqrt3}{2}\right)^2\le0$
Suy ra phương trình $f(x)=0$ luôn có nghiệm.