= Solution
For fixed $a\ne0$ and $B>0$, the rectangle
$$
K=[|a|/2,3|a|/2]\times[-B,B]
$$
is <compact> and stays away from $x=0$. The <function> $f$ in part (i) is continuous there. The <extreme value theorem> makes $|f|$ attain a finite maximum $M_0$ on $K$; choosing $\boxed{M=M_0+1}$ gives the requested strict bound.
The absolute values make this a positive-$x$ rectangle even when $a<0$. Its boundedness is still true, but for a negative initial point the <Picard-Lindelöf theorem> must instead be applied to a rectangle centered at that negative point, as in part (iii).
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