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 .