if d dx f x ( ( )) < , ∀∈ a b . ... ( ) Remark It is very important to note the following fact. It is false to say that if a differentiable function ( ) on I is strictly increasing on I , then ′ > for all x I ∈ .
For instance, consider ∈−∞∞ , ) . It is strictly increasing on ( −∞∞ . To prove this, let a > . Then we have to prove that f a f b > .
For this purpose, we have to prove a > Now, = ( )( b a ab = ( ab = ( > since a > and other terms inside the bracket are > . Hence it is clear that the quadratic expression is always positive (it is equal to zero only if = , which contradicts the condition a ). Therefore the function is y is strictly increasing in ( −∞∞ . But ′ which is equal to zero at x = .
Definition . A stationary point ( )) of a differentiable function f x ( ) is where ′ . Definition . A critical point ( )) of a function f x ( ) is where ′ or does not exist.
Note We State that if ( x,y ) is a Stationary point or Critical Point of f where x from the domain of f is called Stationary number or Critical number Every stationary point is a critical point however all critical points need not be stationary points. As an example, the function f x | | − has a critical point at ( , ) but ( , ) is not a stationary point as the function has no derivative at x = . Example . Prove that the function f x ( ) = is strictly increasing in the interval ( , ) and strictly decreasing in the interval ( , ) −