Let f: A R be a function, then
* f is said to be monotonically increasing on A if x1, x2 A, x1 < x2 f(x1) ≤ f(x2).
* f is said to be strictly increasing on A if x1, x2 A, x1 < x2 f(x1) < f (x2)
* f is said to be monotonically decreasing on A if x1, x2 A, x1 < x2 f(x1) ≥ f(x2)
* f is said to be strictly decreasing on A if x1, x2 A, x1 < x2
* f is said to be monotonic on A if f is either monotonically increasing or decreasing.
Let f be a function defined on a neighbourhood A of a real number "a"
Then f is said to be locally
Increasing at 'a' if
* x A, x < a f(x) < f(a)
* x A, x > a f(x) > f(a)
Decreasing at 'a' if
* x A, x < a f(x) > f(a)
* x A, x > a f(x) < f(a)
Let f be a function defined on a neighbourhood A of a real number 'a'
If f'(a) > 0 then f is increasing at 'a'.
If f' (a) < 0 then f is decreasing at 'a'.
Let f : A
f(a) is also called stationary value of f at 'a' and the point ( a, f(a) ) is called stationary
point (or) turning point of f.
Let f : A R be a function and l f(A). Then 'l' is said to be
The maximum (or) greatest value of f in A if f (x) ≤ l x A
The minimum (or) least value of f in A if f (x) ≥ l x A
Note: A function need not have maximum (or) minimum values in a set.
Let f be a function defined on a neighbourhood A of a real number 'a'
Then f is said to have
Note: f(a) is called relative maximum (or) relative minimum of f at 'a'
Let f be function defined on [a, b]
The maximum of all relative maximum values of f on [a, b] is called absolute maximum of f on [a, b]
The minimum of all relative minimum values of f on [a, b] is called absolute minimum of f on [a, b]
Note: The relative maximum and the relative minimum values of f are called EXTREME Values of f.