= Solution
{c}
The weak equation implies $u''=x^2u-f\in L^2(a,b)$ in <distributions>, hence $u\in H^2(a,b)$. Its nodal interpolant satisfies
$$
\|(u-I_hu)'\|_2\leq\frac h\pi\|u''\|_2,\qquad
\|u-I_hu\|_2\leq\frac{h^2}{\pi^2}\|u''\|_2.
$$
These follow interval by interval: the <interpolation> error vanishes at both endpoints, and its <derivative> has mean zero, so two Poincaré inequalities apply. If $M$ and $\gamma$ are the continuity and <coercivity> constants, <Céa's lemma> gives
$$
\|u-u_h\|_{H^1}\leq\frac M\gamma\|u-I_hu\|_{H^1}
\leq\frac M\gamma\sqrt{\frac{h^2}{\pi^2}+\frac{h^4}{\pi^4}}\,\|u''\|_2
=\boxed{O(h)}.
$$
All constants are independent of the mesh size.
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