Property of f: | Definition: | If differentiable: | If twice differentiable: |
---|---|---|---|

Positive: | f(a) > 0; | ||

Negative: | f(a) < 0; | ||

Increasing: | [f(b) − f(a)] ÷
(b − a) >
0, |
f′(a) > 0; | |

Decreasing: | [f(b) − f(a)] ÷
(b − a) <
0, |
f′(a) < 0; | |

Concave upward: | {[f(c) − f(b)] ÷ (c − b) − [f(b) − f(a)] ÷ (b − a)} ÷
(c − a) >
0, |
[f′(b) − f′(a)] ÷
(b − a) >
0, |
f′′(a) > 0; |

Concave downward: | {[f(c) − f(b)] ÷ (c − b) − [f(b) − f(a)] ÷ (b − a)} ÷
(c − a) <
0, |
[f′(b) − f′(a)] ÷
(b − a) <
0, |
f′′(a) < 0. |

Generally, it's much easier to work with the rightmost condition for every property, but you can't do that if the necessary derivatives don't exist. Even if the function isn't differentiable at all, it still makes sense to say whether or not it's concave upward or downward.

Incidentally, here is some other terminology that you may see for these properties:

- Sometimes people use ≥ and ≤ in place of > and <. If you want to be clear, you can use adverbs: ‘strictly’ for the definitions above (using > and <) or ‘weakly’ for the versions with ≥ and ≤.
- Sometimes people put the word ‘monotone’ in front of ‘increasing’ and ‘decreasing’, even though it really isn't necessary. (However, when people use this word, they are more likely to mean ‹weakly› too, even if they don't say so.)
- Alternatively, if the word ‘monotone’ is used alone, then it means ‹increasing›; the corresponding word for ‹decreasing› is ‘antitone’ (but this word is fairly rare). Again, people who use this terminology are more likely to mean ‹weakly›.
- If the word ‘concave’ is used alone, then it means ‹concave downard›; the corresponding word for ‹concave upward› is ‘convex’ (and this word is extremely common). People who use this terminology are also more likely to mean ‹weakly›.

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