Solution
= Solution
For nodes $0,1$ and $f(x)=e^x$, the cubic <Hermite interpolating polynomial> is
$$
h(x)=1+x+(2e-5)x^2+(3-e)x^3.
$$
Indeed its values at zero and one are $1,e$, and its <derivatives> there are also $1,e$. The <Hermite interpolation error formula> gives
$$
|e^x-h(x)|=\frac{e^\xi}{4!}x^2(1-x)^2
\le\frac e{24}\left(\frac14\right)^2=\boxed{\frac e{384}},
$$
since $0\le\xi\le1$ and $x(1-x)\le1/4$ on $[0,1]$.