- - Applications of Differential Calculus Example . Find the intervals of monotonicity and hence find the local extrema for the function f x ( ) = . We have, ( ) = ( x − , then ′ f ( ) = x − gives x = . The intervals of monotonicity are ( , ) −∞ and ( , ¥ .
Since ′ , for x ∈−∞ , ) the function f x ( ) is strictly decreasing on ( , ) −∞ . As ′ > , for x ∈ ∞ ( , the function f x ( ) is strictly increasing on ( , ¥ . Because ¢ f ( ) changes its sign from negative to positive when passing through x = for the function f x ( ) , it has a local minimum at x = . The local minimum value is f ( ) Example .
Find the intervals of monotonicity and hence find the local extrema for the function f x ( ) = . We have, f x ( ) = , then ′ ( ) = . ′ ( ) ≠ ∀∈ and ′ ( ) x does not exist at x = . Therefore, there are no stationary points but there is a critical point at x = .
Interval (-∞, ) ( , ∞) Sign of ′ ( ) _ Monotonicity strictly decreasing strictly increasing Table . Because ′ ( ) x changes its sign from negative to positive when passing through x = for the function f x ( ) , it has a local minimum at x = .The local minimum value is f ( ) = . Note that here the local minimum occurs at a critical point which is not a stationary point. Example .
Prove that the function f x is increasing on the real line. Also discuss for the existence of local extrema. Since