= Solution
The upper branch leaves the origin horizontally and curves upward, growing like $x^{5/2}$. The lower branch shares the horizontal tangent, rises to its small maximum at $x=16/25$, then crosses the axis at $x=1$ and falls toward $-\infty$. Its <concavity> changes at $x=64/225$, before the maximum. A separate zoom makes those nearby lower-branch features visible.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2-singular-branches.png]
{title=Real branches of the singular curve, with the lower branch enlarged to mark its extrema, inflection, and zeros}
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