Solution
= Solution
\b[No.] On $[-1,1]$, take
$$
f(x)=x^3,\qquad \xi=0.
$$
Then $f'(0)=0$. For every $x_1<0<x_2$,
$$
\frac{f(x_2)-f(x_1)}{x_2-x_1}
=x_2^2+x_1x_2+x_1^2.
$$
This quadratic expression is strictly positive: it equals
$$
\left(x_2+\frac{x_1}{2}\right)^2+\frac34x_1^2.
$$
\b[Thus no chord with endpoints on opposite sides of $\xi$ has slope $f'(\xi)=0$.]