= Solution
The <Lagrange interpolation> polynomials for these nodes are
$$
\ell_1(\tau)=2\tau^2-3\tau+1,\qquad
\ell_2(\tau)=4\tau-4\tau^2,\qquad
\ell_3(\tau)=2\tau^2-\tau.
$$
Their integrals give the <Lobatto IIIA method> tableau
$$
\boxed{\begin{array}{c|ccc}
0&0&0&0\\
1/2&5/24&1/3&-1/24\\
1&1/6&2/3&1/6\\ \hline
&1/6&2/3&1/6
\end{array}.}
$$
To verify its nonlinear order, not just its scalar linear order, set $e=(1,1,1)^T$, $c=Ae$ and $C=\operatorname{diag}(c)$. Direct multiplication verifies the <fourth-order conditions for a Runge-Kutta method>:
$$
b^Te=1,\quad b^Tc=\frac12,\quad b^Tc^2=\frac13,\quad
b^TAc=\frac16,\quad b^Tc^3=\frac14,\quad
b^TCAc=\frac18,\quad b^TAc^2=\frac1{12},\quad
b^TA^2c=\frac1{24}.
$$
Powers of $c$ are componentwise. These eight <Butcher order conditions> establish order at least four for general smooth ODEs. On the <Dahlquist test equation>, elimination of the stages gives
$$
R(z)=\frac{1+z/2+z^2/12}{1-z/2+z^2/12},\qquad
R(z)-e^z=-\frac{z^5}{720}+O(z^6).
$$
A nonzero fifth-order step defect rules out order five. Thus \b[the method has exactly order four].
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