Why does $f(x) = x \cos 4x$ lie below the $x$-axis in ther interval $[\pi/4, \pi/3]$?

Question

How can I explain why function $f(x)$ lies below the $x$-axis in the interval $\left[\dfrac{\pi} 4, \dfrac{\pi} 3\right]$ where $f(x) = x \cos 4x$?

Answer 1

If $x$ is in the interval $\left[\dfrac{\pi} 4, \dfrac{\pi} 3\right]$ then $4x$ is between $\pi$ and $\dfrac{4\pi} 3$, which puts $4x$ in the third quadrant. Hence $\cos 4x$ is negative and so is $x \cos 4x$ and its graph is below the $x$-axis.