Corollary

Tag 0852, permanent. Cite it as [Bourbaki, Tag 0852].

Functions of a Real Variable, chapter I, § 3. DERIVATIVES OF HIGHER ORDER

If $f$ is a finite real function with a derivative of order $n + 1$ on I, and if $m \leq f^{(n+1)}(x) \leq M$ on I, then for all $x \geq a$ in I one has

m \frac{(x - a)^{n+1}}{(n + 1)!} \leq r_n(x) \leq M \frac{(x - a)^{n+1}}{(n + 1)!}

and the second term cannot be equal to the first (resp. to the third) unless

f^{(n+1)}

is constant and equal to

m

(resp.

M

) on the interval

[a, x]

The proof proceeds in the same way, but applying th. 1 of I, p. 14.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

I, p. 14