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) . xa
xa
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) . xa
xa
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
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