32
This problem illustrates the fact that two functions can be very close: $|f(x)-g(x)|<\epsilon$
for all $x\in [0,1]$, but their derivatives can still be far apart, $|f'(x)-g'(x)|>c$ for some
constant $c>0$.
In our case, let $x=a(t),y=b(t),0\le t\le 1$ and $x=c(t),y=d(t), 0\le t\...