= Solution
{c}
Under the stated piecewise smooth hypotheses, the <Dirichlet convergence theorem for Fourier series> gives the average of the two one-sided limits of the corresponding <periodic extension>. The <even extension> is continuous at both endpoints, so the <Fourier cosine series> converges to
$$
\boxed{f(x)\quad\text{for every }x\in[0,L].}
$$
The <odd extension> agrees with $f$ on $(0,L)$, but its left and right limits are $-f(0),f(0)$ at zero and $f(L),-f(L)$ at the join $L$. Therefore the <Fourier sine series> converges to
$$
\boxed{\begin{cases}f(x),&0<x<L,\\0,&x=0\text{ or }x=L.\end{cases}}
$$
Indeed every <partial sum> of the <Fourier sine series> is already zero at both endpoints. The endpoint values agree with $f$ only when the corresponding <boundary values> vanish.
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