# Difference between revisions of "Positive derivative implies increasing"

From Calculus

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===On an open interval=== | ===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</math> is of the form <math>\! (a,b)</math>, <math>(a,\infty)</math>, <math>(-\infty,b)</math>, or <math>(-\infty,\infty)</math>). Suppose the [[derivative]] of <math>f</math> exists and is positive everywhere on <math>I</math>, i.e., <math>f'(x) > 0</math> for all <math>x \in I</math>. Then, <math>f</math> is an [[fact about::increasing function]] on <math>I</math>, i.e.: | + | 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</math> is of the form <math>\! (a,b)</math>, <math>(a,\infty)</math>, <math>(-\infty,b)</math>, or <math>(-\infty,\infty)</math>). Suppose the [[derivative]] of <math>f</math> exists and is positive everywhere on <math>I</math>, i.e., <math>\! f'(x) > 0</math> for all <math>x \in I</math>. Then, <math>f</math> is an [[fact about::increasing function]] on <math>I</math>, i.e.: |

<math>x_1,x_2 \in I, x_1 < x_2 \implies f(x_1) < f(x_2)</math> | <math>x_1,x_2 \in I, x_1 < x_2 \implies f(x_1) < f(x_2)</math> |

## Latest revision as of 16:56, 13 December 2011

## 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 . There exists such that and equals this difference quotient. | Fact (1) | , is defined and continuous on an interval containing , differentiable on the interior of the interval. | [SHOW MORE] | |

2 | The difference quotient is positive. | is positive for all in the interior of . | Step (1) | [SHOW MORE] | |

3 | Step (2) | [SHOW MORE] |