Here you will learn definitions and concepts of local and absolute maxima and minima class 12.
Let’s begin –
Maxima and Minima Class 12
Local Maxima
A function f(x) is said to have local maxima at x = a if there exist a neighbourhood (a – h, a + h) – {a} such that
f(a) > f(x) for all x \(\in\) (a – h, a + h) – {a}
Local Minima
A function f(x) is said to have local minima at x = a if there exist a neighbourhood (a – h, a + h) – {a} such that
f(a) < f(x) for all x \(\in\) (a – h, a + h) – {a}
Absolute Maxima (Global Maxima)
A function f has an absolute maxima (or global maxima) at c if f(c) \(\ge\) f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of on D.
Absolute Minima (Global Minima)
A function f has an absolute minima (or global minima) at c if f(c) \(\le\) f(x) for all x in D and the number f(c) is called the minimum value of on D. The maximum and minimum values of f are called the extreme values of f.
Note :
(i) the maximum & minima values of a function are also known as local/relative maxima or local/relative minima as these are greatest & least values of the function relative to some neighbourhood of the point in question.
(ii) the term ‘extrema’ is used both for maxima or minima.
(iii) a maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.
(iv) a function can have several extreme values & a local minimum value may even be greater than a local maximum value.
(v) local maximum & local minimum values of a continuous function occur alternatively & between two consecutives local maximum values there is a local minimum value & vice versa.
