= Solution
At $x=0$, $g''(0)=36>0$, so the <second-derivative test> gives a strict local <minimum>. In fact $g'(x)=4x(x-3)^2$ is negative for $x<0$ and positive for $x>0$ except at $x=3$, so $g(0)=5$ is the global <minimum> as well.
At $x=3$, the <second-derivative test> is inconclusive because $g''(3)=0$. But $g''(x)=12(x-1)(x-3)$ changes from negative to positive across $3$, while $g'(3)=0$. Hence this is a <stationary inflection point>. Equivalently, the leading nonconstant term $4(x-3)^3$ in the <Taylor polynomial> changes sign across $3$. The paper explicitly uses “saddle point” to mean an <inflection point>, so in its terminology the classification is
$$
\boxed{x=0:\text{ minimum};\qquad x=3:\text{ saddle point};\qquad\text{no maxima}.}
$$
Here the “saddle point” is a one-dimensional <stationary inflection point>, rather than the usual multivariable use of <saddle point of a scalar function>.<Discrete mathematics, continuous mathematics, and probability resit examination by the University of Oxford (2015)>[Page 8, Question 6(b)(iii), explicit definition of “saddle point”]{ref}
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