Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 5 / Solution

Factorization gives $y=(x-2{)}^2(x+1)$, so the zeros are $x=-1$ and $x=2$, with a double root at two. The derivatives are $y^{\prime }=3x(x-2)$ and $y^{\prime \prime }=6(x-1)$. ThusThe second derivative changes sign at one, confirming the inflection point. The curve rises from $-\infty$, crosses at $-1$, reaches its local maximum, falls to touch the axis at two, then rises to $+\infty$. Neither local extremum is a global extremum on the whole real line.