Exercise 10, § 3

Functions of a Real Variable, I, 42

Let $g$ be an odd real function defined on a neighbourhood of 0, and 5 times differentiable on this neighbourhood. Show that

g(x) = \frac{x}{3} \left( g'(x) + 2g'(0) \right) - \frac{x^5}{180} g^{(5)}(\xi) \quad (\xi = \theta x, \quad 0 < \theta < 1)

(same method as in exerc. 9).

Deduce that if $f$ is a real function defined on $[a, b]$ and 5 times differentiable on this interval, then

f(b) - f(a) = \frac{b - a}{6} \left[ f'(a) + f'(b) + 4f'\left( \frac{a + b}{2} \right) \right] - \frac{(b - a)^5}{2880} f^{(5)}(\xi)

with

a < \xi < b

("Simpson's formula").

Solution

No solution has been written yet.