Consider a function $f(x)$ defined on the domain $x \in[0,1]$. Find constants $\alpha, \beta$, so that for any fixed $\xi \in[0,1]$,

$$f^{\prime \prime}(\xi)=\alpha f(0)+\beta f^{\prime}(0)+\gamma f(1)$$

is exactly satisfied for polynomials of degree less than or equal to two.

By using the Peano kernel theorem, or otherwise, show that

$$\begin{aligned} f^{\prime}(\xi)-f^{\prime}(0)-\xi(\alpha f(0)&\left.+\beta f^{\prime}(0)+\gamma f(1)\right)=\int_{0}^{\xi}(\xi-\theta) H_{1}(\theta) f^{\prime \prime \prime}(\theta) d \theta \\ &+\int_{0}^{\xi} \theta H_{2}(\theta) f^{\prime \prime \prime}(\theta) d \theta+\int_{\xi}^{1} \xi H_{2}(\theta) f^{\prime \prime \prime}(\theta) d \theta \end{aligned}$$

where $H_{1}(\theta)=1-(1-\theta)^{2} \geqslant 0, H_{2}(\theta)=-(1-\theta)^{2} \leqslant 0$. Thus show that

$$\left|f^{\prime}(\xi)-f^{\prime}(0)-\xi\left(\alpha f(0)+\beta f^{\prime}(0)+\gamma f(1)\right)\right| \leqslant\left.\frac{1}{6}\left(2 \xi-3 \xi^{2}+4 \xi^{3}-\xi^{4}\right)|| f^{\prime \prime \prime}\right|_{\infty} .$$