Confusion with the Taylor's Inequality

[yupenn]

Taylor's Inequality states:
if |f n+1(x)|<=M
then
|Rn(x)|<=M*|x-a|^(n+1)/(n+1)!
however,
Rn(x)=f n+1(x)*|x-a|^(n+1)/(n+1)!+...
it seems |Rn(x)|>=M*|x-a|^(n+1)/(n+1)! when |f n+1(x)|=M

the inequality is wrong??

 

[HallsofIvy]

You seem to be assuming that all derivatives of f are positive which is NOT generally true.