Positive derivative implies increasing
From Calculus
Contents
Statement
On an open interval
Suppose is a function on an open interval
that may be infinite in one or both directions (i..e,
is of the form
,
,
, or
). Suppose the derivative of
exists and is positive everywhere on
, i.e.,
for all
. Then,
is an increasing function on
, i.e.:
On a general interval
Suppose is a function on an interval
that may be infinite in one or both directions and may be open or closed at either end. Suppose
is a continuous function on all of
and that the derivative of
exists and is positive everywhere on the interior of
, i.e.,
for all
other than the endpoints of
(if they exist). Then,
is an increasing function on
, i.e.:
Related facts
Similar facts
- Zero derivative implies locally constant
- Negative derivative implies decreasing
- Nonnegative derivative that is zero only at isolated points implies increasing
- Increasing and differentiable implies nonnegative derivative
Facts used
Proof
General version
Given: A function on interval
such that
for all
in the interior of
and
is continuous on
. Numbers
with
.
To prove:
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Consider the difference quotient ![]() ![]() ![]() ![]() |
Fact (1) | ![]() ![]() ![]() ![]() |
[SHOW MORE] | |
2 | The difference quotient ![]() |
![]() ![]() ![]() |
Step (1) | [SHOW MORE] | |
3 | ![]() |
![]() |
Step (2) | [SHOW MORE] |