Maxima and Minima Class 12

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.

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