Exercise 6.7.11

Question

Assume that $f$ has a continuous derivative on $[a, b]$. Show that there exists a polynomial $p(x)$ such that
$$
|f(x)-p(x)|<\epsilon \quad 	ext { and } \quad\left|f^{\prime}(x)-p^{\prime}(x)ight|<\epsilon
$$
for all $x \in[a, b]$.

Ulisse Mini · Accepted Answer

Let $q(x)$ be a polynomial which uniformly approximates $f'(x)$ to within $\frac{\epsilon}{(b-a)}$, and $p(x)$ to be the antiderivative of $q(x)$ which passes through $(a, f(a))$. Let $e(x) = f(x) - p(x)$ be the error between $f$ and $p$, and note that $\left|e'(x)\right| < \frac{\epsilon}{b-a}$.

By the Mean Value Theorem, for any $x \in [a, b]$,
$$\left|e(x) - e(a)\right| = \left|(x-a)e'(c)\right| < (b-a) \frac{\epsilon}{(b-a)} = \epsilon$$
for some $c \in [a,b]$, completing the proof.

Exercise 6.7.11

Answers

Comments

Exercise 6.7.11

Answers

Comments

Add answer