f x ( ) = + . The given function is defined and is differentiable at all x ∈−∞∞ ) . As ( ) = + x , we have ′ f ( ) = − ) . - - Applications of Differential Calculus The stationary numbers are given by − ) = that is x = .
Hence the intervals of monotonicity are ( , ) −∞ and ( , ¥ . On the interval ( , ) −∞ the function strictly increases because ′ > in that interval. The function f x ( ) strictly decreases in the interval ( , ¥ because ′ in that interval. Since ′ f ( ) changes from positive to negative when passing through x = , the first derivative test tells us there is local maximum at x = and the local maximum value is f ( ) = .
Example . Find the intervals of monotonicity and local extrema of the function f x ( ) = + . The given function is defined and differentiable at all x ∈−∞∞ ) , As ( ) = ′ f ( ) = ) . The stationary numbers are given by − x = that is x = ± .
Hence the intervals of monotonicity are ( ),( , ) −∞− and ( , ¥ . Interval (-∞, - ) (- , ) ( , ∞) Sign of ′ ( ) _ _ Monotonicity strictly decreasing strictly increasing strictly decreasing Table . Therefore, f x ( ) strictly increasing on ( −∞− and ( , ¥ , strictly decreasing on ( , ) - . Since ′ f ( ) changes from negative to positive when passing through x = − , the first derivative test tells us there is a local minimum at x = − and the local minimum value is f ( = − .