i have some problems with the following task:
Let $f\in C^2([a,b])$ and s the interpolating linear spline of f with the grid points $x_i=x_0+ih, i=0,...,n$ and $h=\frac{1}{n}(b-a)$. Proof for every point $x\in(x_i,x_{i-1})$, that $|f'(x)-s'(x)|\leq\frac{h}{2} \max_{\xi\in[a,b]}|f''(\xi)|$.
My idea was to look at the derivative of s. Then $s'(x)=\dfrac{f(x_i)-f(x_{i-1})}{x_i-x_{i-1}}$, but i don't know how to use it. In the last task i had to proof the equation $|f(x)-s(x)|\leq\frac{h^2}{8} \max_{\xi\in[a,b]}|f''(\xi)|$. Maybe I should use this in my proof? I don't have any idea.
Maybe someone could help me and give some hints how to go on ? :)