′ = − ≥ and zero at the points x π , and hence the function is increasing on the real line. Since there is no sign change in ′ f ( ) when passing through x π , by the first derivative test there is no local extrema. Example . Discuss the monotonicity and local extrema of the function x x log( −+ > − .
Fig. . - - We have, ( ) = log( −+ Therefore, ′ f ( ) = Hence, ′ f ( ) is −< > > when when when Therefore f x ( ) is strictly increasing for x > and strictly decreasing for x < . Since ′ f changes from negative to positive when passing through x = , the first derivative test tells us there is a local minimum at x = which is f ( ) Example .
Find the intervals of monotonicity and local extrema of the function f x + . The given function is defined and is differentiable at all x ∈ ∞ ( , ( ) = x + . Therefore ′ f ( ) = log + + The stationary numbers are given by + log x = . That is x = e − .
Hence the intervals of monotonicity are ( , e − and ( e − ∞ At x ′ = −< ( , ), and hence in the interval ( , e − the function is strictly decreasing. At x ∞ ′ = > ), and hence strictly increasing in the interval ( e − ∞ . 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 it is f e = − . Example .
Find the intervals of monotonicity and local extrema of the function