Does there exist a quadratic function $f(x)$ such that the conditions (1) when $|x| \leqslant 1$, $|f(x)| \leqslant 1$; (2) $|f(2)| \geqslant 7$ hold simultaneously?
Solution: Existence, such as $f(x)=2 x^{2}-1$. It is easy to prove that it satisfies the condition.