Optimizing Polynomial Approximations for C2 Functions on Closed Intervals

[Demon117]

1. Suppose that f:R-->R is of class C². Given b>0 and a positive number $$\epsilon$$, show that there is a polynomial p such that

|p(x)-f(x)|<$$\epsilon$$, |p'(x)-f'(x)|<$$\epsilon$$, |p"(x)-f"(x)|<$$\epsilon$$ for all x in [0,b].

The Attempt at a Solution

First I choose a polynomial q that approximates f''. If |q - f''|<$$\eta$$ throughout [0,b], and if p is the polynomial such that p''=q, p(0)=f(0), and p'(0)=f'(0), then I come to this question: How big can |p' - f'| and |p - f| be in terms of $$\eta$$ and b? I think I am thinking about this correctly, but I cannot come to a conclusion. Should I use the definition of the derivative namely:

For all $$\epsilon$$>0, there is a $$\delta$$>0, such that when 0<|t-x|<$$\delta$$, this guarantees that |(p(t)-p(x))/(t-x) - L|<$$\epsilon$$; where L is the derivative at x.

 

[ystael]

Observe that $$(p' - f')' = p'' - f''$$; what does this tell you about the accumulated error in approximating $$f'$$ by $$p'$$, as you move across the interval from $$0$$ to $$b$$?