Quadratic equations

If we desire $|ax^2 + bx + c| \le 1$ for all $x \in [0,1]$ , then $-1 \le ax^2 + bx + c \le 1.$ At $x = 0$ , we obtain $-1 \le c \le 1 \Rightarrow |c| \le 1.$ At $x = 1,$ we now have $-1 \le a + b + c \le 1$ , and if $|c| \le 1$ then we require $a + b = 0 \Rightarrow b = -a; a \ne 0$ .

Now, let us first examine the quadratic equation $ax^2 - ax + 1 = a(x - \frac{1}{2})^2 + (1 - \frac{a}{4}).$ The minimum(maximum) value of this parabola must satisfy $-1 \le 1 - \frac{a}{4} \le 1 \Rightarrow 0 \le a \le 8$ , or $a \in (0,8]$ since $a \ne 0.$ Also, let us examine $ax^2 - ax - 1 = a(x - \frac{1}{2})^2 - (1 + \frac{a}{4}),$ which requires $-1 \le -1 - \frac{a}{4} \le 1 \Rightarrow -8 \le a \le 0$ , or $a \in [-8,0)$ . In both cases, $|a| \le 8$ , which in turn implies $b = -a \Rightarrow |b| \le 8.$

Hence the maximum value $|a| + |b| + |c| \le k$ equals $|8| + |8| + |1| = 17 \le k$ , or $\boxed{k = 17}.$