increasing function or a decreasing function. Derivative test for increasing and decreasing function Theorem: . (Without Proof) Let f ( x ) be a continuous function on [ a , b ] and differentiable on the open interval ( a , b ), then (i) f ( x ) is increasing in [ a , b ] if ′ ( ) ≥ (ii) f ( x ) is decreasing in [ a , b ] if ′ ( ) ≤ Remarks: (i) f ( x ) is strictly increasing in ( a , b ) if ′ ( ) > for every x a b ∈ ( (ii) f ( x ) is strictly decreasing in ( a , b ) if ′ ( ) < for every x a b ∈ ( (iii) f ( x ) is said to be a constant function if ′ ( ) = . .
Stationary Value of a function Let f ( x ) be a continuous function on [ a , b ] and differentiable in ( a , b ). f ( x ) is said to be stationary at x = a if f ' ( a )= . - - Applications of Differentiation The stationary value of f ( x ) is f ( a ). The point ( a , f ( a ) ) is called stationary point.
Y y = f ( x ) P, Q, R are called Stationary Points P Q Increasing > dy dx > Decreasing > dy dx < Decreasing > Increasing > dy dx > dy dx = Tangent dy dx = Tangent dy dx = Tangent o Fig: . In figure . the function y = f ( x ) has stationary at x = a , x = b and x = c . At these points, dy dx = .
The tangents at these points are parallel to x – axis. NOTE By drawing the graph of any function related to economics data, we