Non-rigorous argument:Originally Posted by nath_quam
Consider the related function:
for any small eps>0. This has three roots all close to x=1, and in some
appropriate sense the average slope between x=1-eps and x=1+eps
is zero, so as eps goes to zero we generate a family of curves all of which
have an "average" slope of 0 near x=1, so it is reasonable to expect that
the slope of p(x) is 0 at x=1.
(I think I would need Roll's theorem (and the nested interval theorem?) to
make this rigorous, and your teacher would almost certainly forbid THAT).