Monotonic conditions. Extremum of function

Theorem 1. Let be continuous on [a, b] and differentiable on (a, b). is a constant

function if and only if for all .

Definition 1. A function is said to be monotonic increasing (resp. monotonic decreasing)

or simply increasing (resp. decreasing) on an interval if and only if , if

then (resp. , if , then ).

Definition 2. A function is said to be strictly increasing (resp. strictly decreasing) on an

interval if and only if , if then (resp. , if ,

then ).

Theorem 2. Let be continuous on [a, b] and differentiable on (a, b). Then

(a) if is strictlyincreasing on [a, b]; and

(b)if is strictly decreasing on [a, b].