Given a function $f(x)$, that is differentiable at least three times for all $x \in [0,1]$, define the associated quadratic polynomial $g(x)$ by the properties $$f(0)=g(0),$$ $$f(1) = g(1),$$ $$f'(0) = g'(0).$$ Give a formula for $g(x)$ and derive a Lagrange-type formula for the remainder $r(x)$, when $g(x)$ is used to approximate $f(x)$ in the interval $(0,1)$
I'm fine with finding $g(x)$ (or so I think) giving me $g(x) = (f(1) - f'(0)-f(0))x^2 + f'(0)x+f(0)$ by simply comparing coefficients however as this does not seem to be in the form $\sum\frac{(x-a)^kf^k(a)}{k!}$ of sort so I'm completely unsure with how to proceed with finding the remainder.