Iit mathematics 2

Page 1

12. MAXIMA & MINIMA (1)

Basic definitions: (i) Stationary points: For a given function f(x), stationary points are those locations of x (let it be 'x0') for which f'(x0) = 0, and functioning value f(x0) is termed as the stationary value of function. (ii) Critical points: For a given function f(x), critical points are those locations of x(let it be 'x') in the domain of f(x) for which f'(x0) = 0 or f'(x0) doesn't exist.

Note: At critical points for a function, local maxima, local minima

or point of inflection may

exist (iii) Points of extremum: critical points of a function at which either maxima or minima exists is termed as points of extremum. (iv) Point of inflection: critical locations of function at which neither maxima nor minima exists is termed as point of in flection, it is also known as point of no extremum. (v) Local maxima/regional maxima: A function f(x) is said to have a local maximum at critical point x= a if f(a)  f(x)  x  (a  h,a + h) , where h is a very small positive arbitrary number. If x = a is the point of local maxima, then f(a)  lim f(x) and f(a)  lim f(x) . xa

xa

following figure illustrates the point of local maxima in different situations:

(vi) Local minima/regional minima: A function f(x) is said to have local minimum at critical point x = a if f(a)  f(x)  x  (a  h, a + h), where h is a very small positive arbitrary number. If x = a is the point of local minima, if then f(a)  lim f(x) and f(a)  lim f(x) .   xa

xa

following figure illustrates the point of local minima in different situations:

ma r a h S Note: . 82 K . 6 L 7 7 in the (i) local maxima of a function at x = a is the largest value function 8nearby 5 Er.of 8 02only 1 0 neighbourhood of critical point x = a. 9 8015 (ii) local minima of a function at x = a is the smallest value of function 39 only in the nearby 8 neighbourhood of critical point x = a. Mathematics Concept Note IIT-JEE/ISI/CMI

Page 97


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.