- #1
Demon117
- 162
- 1
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].
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.
|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.