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**mathematicalbagpiper** Suppose that $\displaystyle f:\mathbb{R}\longrightarrow\mathbb{R}$ is of class $\displaystyle C^2$. Given $\displaystyle b>0$ and a positive number $\displaystyle \epsilon$, show that there exists a polynomial $\displaystyle p$ such that:

$\displaystyle |p(x)-f(x)|<\epsilon$

$\displaystyle |p'(x)-f'(x)|<\epsilon\\$

$\displaystyle |p''(x)-f''(x)|<\epsilon$

For all $\displaystyle x\in[0,b]$.

The hint given was to choose a polynomial $\displaystyle q$ that approximates $\displaystyle f''$, and suppose that $\displaystyle |q-f''|<\eta$ throughout the interval $\displaystyle [0,b]$. Now say that $\displaystyle p''=q, p(0)=f(0), p'(0)=f'(0)$, and to bound $\displaystyle |p'-f'|$ and $\displaystyle |p-f|$ in terms of both $\displaystyle \eta$ and $\displaystyle b$.

So I'm thinking to choose that $\displaystyle q$ and try to find an antiderivative, and then find an antiderivative of that such that the other two conditions hold, but I'm having trouble seeing how those distances all relate to one another.