3. If real numbers $a, b, c$ make the quadratic function
$$
f(x)=a x^{2}+b x+c
$$

satisfy $|f(x)| \leqslant 1$ for $0 \leqslant x \leqslant 1$, then the maximum value of $2|a|+$ $\sqrt{2}|b|+|c|$ is . $\qquad$
3. $17+8 \sqrt{2}$.

Let $x=1,0.5,0$, we get
$$
c=f(0) \text {, }
$$
$$
\begin{array}{l}
b=-f(1)+4 f(0.5)-3 f(0), \\
a=2 f(1)-4 f(0.5)+2 f(0) .
\end{array}
$$

From the absolute value inequality, we know
$$
\begin{array}{l}
2|a|+\sqrt{2}|b|+|c| \\
\leqslant(4+\sqrt{2})|f(1)|+(8+4 \sqrt{2})|f(0.5)|+ \\
(5+3 \sqrt{2})|f(0)| \\
\leqslant 17+8 \sqrt{2} \text {. } \\
\end{array}
$$