> for each x ∈ ( a , b ) (b) f is decreasing in [ a,b ] if f ′ ( x ) < for each x ∈ ( a , b ) (c) f is a constant function in [ a,b ] if f ′ ( x ) = for each x ∈ ( a , b ) Strictly Increasing function (i) Neither Increasing nor Decreasing function (iii) Strictly Decreasing function (ii) Proof (a) Let x , x ∈ [ a , b ] be such that x < x . Then, by Mean Value Theorem (Theorem in Chapter ), there exists a point c between x and x such that f ( x ) – f ( x ) = f ′ ( c ) ( x – x ) i.e. f ( x ) – f ( x ) > (as f ′ ( c ) > (given)) i.e. f ( x ) > f ( x ) Thus, we have ), for all [ , ] f x f x x x a b Hence, f is an increasing function in [ a,b ].
The proofs of part (b) and (c) are similar. It is left as an exercise to the reader. Remarks There is a more generalised theorem, which states that if f ¢( x ) > for x in an interval excluding the end points and f is continuous in the interval, then f is increasing. Similarly, if f ¢( x ) < for x in an interval excluding the end points and f is continuous in the interval, then f is decreasing.
Example Show that the function f given by f ( x ) = x – x + x , x ∈ R is increasing on R . Solution Note that f ′ ( x ) = x – x + = ( x – x +