The point x = a is called a point of minimum of the function f( x) and f( a) is known as the minimum value or the least value or the absolute minimum value of f( x).
Properties of Maxima and Minima
1. If f( x) is continuous function in its domain, then at least one maxima and one minima must lie between two equal values of x.
2. Maxima and minima occur alternately, i. e., between two maxima there is one minima and vice-versa.
3. If f( x) → ∞ as x → a or b and f ‗( x) = 0 only for one value of x( sayc) between a and b, then f( c) is necessarily the minimum and the least value.
4. If f( x) → p-∞ as x → a or b and f( c) is necessarily the maximum and the greatest value.
Important Points to be Remembered
1. If f( x) be a differentiable functions, then f ‗( x) vanishes at every local maximum and at every local minimum.
2. The converse of above is not true, i. e., every point at which f‘( x) vanishes need not be a local maximum or minimum. e. g., if f( x) = x 3 then f ‗( 0) = 0, but at x = 0. The function has neither minimum nor maximum. In general these points are point of inflection.
3. A function may attain an extreme value at a point without being derivable there at, e. g., f( x) = | x | has a minima at x = 0 but f '( 0) does not exist.
4. A function f( x) can has several local maximum and local minimum values in an interval. Thus, the maximum and minimum values of f( x) defined above are not necessarily the greatest and the least values of f( x) in a given interval.