Sharp central-secant derivative error
= Sharp central-secant derivative error
{title2=$|f'(x)-(f(x+h)-f(x-h))/(2h)|\le h^2\|f'''\|_\infty/6$}
For $f\in C^3[x-h,x+h]$, the <Peano kernel> for the error of the central secant slope has one sign. Integrating its absolute value gives the displayed constant $h^2/6$, and a cubic polynomial attains it. At $x=1$, $h=1$, the unnormalized order-two kernel is $-\theta^2/2$ on $[0,1]$ and $-(2-\theta)^2/2$ on $[1,2]$; the integral representation includes the factor $1/2!$.