Positive derivative implies increasing

From Calculus
Revision as of 20:42, 20 October 2011 by Vipul (talk | contribs) (Created page with "==Statement== ===On an open interval=== Suppose <math>f</math> is a function on an open interval <math>I</math> that may be infinite in one or both directions (i..e, <math>I</m...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Statement

On an open interval

Suppose f is a function on an open interval I that may be infinite in one or both directions (i..e, I is of the form (a,b), (a,\infty), <math>(-\infty,b), or (-\infty,\infty)). Suppose the derivative of f exists and is positive everywhere on I, i.e., f'(x) > 0 for all x \in I. Then, f is an increasing function on I, i.e.:

x_1, x_2 \in I, \qquad x_1 < x_2 \implies f(x_1) < f(x_2)

Facts used

  1. Lagrange mean value theorem

Proof

Fill this in later