## How do you know if a Function is Increasing or Decreasing?

### Increasing and decreasing functions

**Definition:**

(1) A function f is said to be an increasing function in ]a,b[, if x_{1} < x_{2} ⇒ f(x_{1}) < f(x_{2}) for all x_{1}, x_{2} ∈ ]a,b[.

(2) A function f is said to be a decreasing function in ]a,b[, if x_{1} < x_{2} ⇒ f(x_{1}) < f(x_{2}), ∀ x_{1}, x_{2} ∈ ]a,b[.

f(x) is known as non-decreasing if f’(x) ≥ 0 and non-increasing if f’(x) ≤ 0.

**Monotonic function:** A function f is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

We summarize the results in the table below :

f’(a_{1}) | f’’(a_{1}) | f’’’(a_{1}) | Behaviour of f at a_{1} |

+ | Increasing | ||

– | Decreasing | ||

0 | + | Minimum | |

0 | – | Maximum | |

0 | 0 | ? | |

0 | 0 | ± | Inflection |

0 | 0 | ? |

- Blank space indicates that the function may have any value at a
_{1}. - Question mark indicates that the behaviour of f cannot be inferred from the data.

### Properties of monotonic functions

- If f(x) is a strictly increasing function on an interval [
*a, b*], then f^{−1}exists and it is also a strictly increasing function. - If f(x) is a strictly increasing function on an interval [
*a, b*] such that it is continuous, then f^{−1}is continuous on [f(a), f(b)]. - If f(x) is continuous on [
*a, b*] such that f’(c) ≥ 0 [f’(c) > 0] for each c ∈ (*a*,*b*), then f(x) is monotonically (strictly) increasing function on [*a*,*b*]. - If f(x) is continuous on such that f’(c) ≤ 0 [f’(c) > 0] for each c ∈ (
*a*,*b*), then f(x) is monotonically (strictly) decreasing function on [*a*,*b*]. - If f(x) and
*g*(*x*) are monotonically (or strictly) increasing (or decreasing) functions on [*a*,*b*], then is a monotonically (or strictly) increasing function on [*a*,*b*]. - If one of the two functions f(x), g(x) is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, then
*gof*(*x*) is strictly (monotonically) decreasing on [*a*,*b*].

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