= Solution
The <tangent> has poles at $\pi/2$ and $3\pi/2$, so these are vertical <asymptotes> of $x\tan x$. The zeros on the given interval are $0,\pi,2\pi$. Near zero, $x\tan x=x^2+O(x^4)$, so the graph starts with a horizontal <tangent line>.
Each of its three branches is increasing. Indeed
$$
f'(x)=\frac{x+\sin x\cos x}{\cos^2x},
$$
and the numerator starts at zero with <derivative> $2\cos^2x\geq0$, so it is positive for $x>0$. The first branch rises from zero to $+\infty$, the middle branch rises from $-\infty$ through $(\pi,0)$ to $+\infty$, and the last rises from $-\infty$ to $(2\pi,0)$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1-tangent-circle.png]
{title=The three increasing tangent-product branches and their intersections with the right half of the circle}
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