Finding the error interpolating function.

Question

While looking at the video I've linked at the end of this question about the error formula for interpolating functions, I got confused at the following example $$if n=1, x_0 =a, x_1 =b , b>a $$ find an upper bound for the error. Recall $ e(x) = ((f^{n+1}(c))/(n+1)!)(x-x_0)...(x-x_n) $ The step I'm confused is the following:$$\max_{a<=x<=b}|(x-a)(x-b)|= ...=(b-a)^2/4$$ How is that true?

Answer 1

Well, you just need to solve ((x-a)*(x-b))' = 0 and you'll get x=(a+b)/2 and function value -(b−a)(b-a)/4