= Solution
The <Peano kernel theorem> says that if a continuous linear functional $L$ annihilates polynomials of degree below $m$, then
$$
L(f)=\int_a^bK(t)f^{(m)}(t)\,dt,
\qquad
K(t)=L\left(\frac{(x-t)_+^{m-1}}{(m-1)!}\right).
$$
Here $m=3$ and
$$
L(f)=f''(0)-f(-1)+2f(0)-f(1).
$$
Applying $L$ to $(x-t)_+^2/2$ gives
$$
K(t)=
\begin{cases}
\frac12(1+t)^2,&-1\leq t<0,\\
-\frac12(1-t)^2,&0\leq t\leq1.
\end{cases}
$$
Hence
$$
|L(f)|
\leq\|f'''\|_\infty\int_{-1}^1|K(t)|\,dt
=\frac13\|f'''\|_\infty.
$$
The smallest candidate is therefore
$$
\boxed{c=\frac13}.
$$
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