Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 18

Along a differentiable graph satisfying the implicit equation, the multivariable chain rule givesThus, wherever $F_y\ne 0$,The implicit function theorem supplies such a local graph under that nonvanishing condition. At a singular point both numerator and denominator vanish, so this division cannot determine its tangent.

For the specified polynomial,The first equation $F_y=0$ forces $y=x^2$; substitution into $F_x=0$ then gives $x=0$, and hence $y=0$. It also satisfies the curve equation. Therefore the unique singular point is

A real square on the left requires $x^5\ge 0$, so $x\ge 0$. Taking the two square roots gives exactly the two real graph branchesThey meet only at the origin. The local coordinate change $Y=y-x^2$ turns the equation into $Y^2=x^5$, the plane cusp of type (2,5); both branches share the tangent $y=0$ there.

For $x>0$, differentiate the two explicit branches:The plus branch is strictly increasing and strictly convex for $x>0$. Its only zero is $x=0$, also its one-sided global minimum; it has no interior extremum or inflection point.

The minus branch rises initially, then has $y_{-}^{\prime }=0$ at $\sqrt{x}=4/5$, changing from positive to negative. Its second derivative changes sign at $\sqrt{x}=8/15$. Its zeros follow from $y_{-}=x^2(1-\sqrt{x})$. ConsequentlyThe origin is a one-sided local minimum of both graph branches, because their domain ends there; it is not an interior critical point. The minus branch has no global minimum since it tends to $-\infty$ as $x\to \infty$. Its maximum and inflection heights, useful for the sketch, are $256/3125$ and $28672/759375$, respectively.

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.