Working Rule For Finding Maxima And Minima

(a) First Derivative test

To check the maxima or minima at x = a, where f ′(a) = 0.

(I) If f ′(x) > 0 at x < a and f ′(x) < 0 at x > a the sign of f ′(x) changes
from +ve to −ve then f(x) has a local maximum at x = a.

(II) If f ′ (x) < 0 at x < a and f ′(x) > 0 at x > a the sign of f ′(x) changes
from −ve to +ve then f(x) has a local minimum at x = a.

(III) If the sign of f ′(x) does not change then f(x) has neither local maximum or minimum at x = a and the point is said to the point of inflexion.

(b) Second Derivative Test

I If f ″(a) < 0 and f ′(a) = 0 then a is a point of a local maximum

II If f″(a) > 0 and f ′(a) = 0 then a is the point of local minimum.

III If f ″(a) = 0 and f ′(a) = 0 then further differentiate and obtain f ′″(a).

IV If f ′(a) = f ″(a) = f″′(a) = …. = f n - 1(a) = 0 and fn(a) ≠ 0

If n is odd then f(x) has neither local maximum nor local minimum at x = a and the point is that of inflexion.

If n is even then if  fn (a) < 0 then f(x) has a local maximum at x = a and if
fn(a) > 0 then f(x) has a local minimum at x = a.

It may be pointed out in this connection that the greatest and least value of continuous function f(x) on the interval [a, b] is attained either at the critical points or at the end points of the interval. To find the greatest value of the function or (least value)
we have to compute its values at all the critical points on the interval [a, b] and the values f(a), f(b) of the function at the end points of the interval and choose the greatest (least) out of them.