derivative test, the function f x ( ) has local minimum. But at x = , ′′ f ( ) . That is the second derivative test does not give any information about local extrema at x = . Therefore, we need to go back to the first derivative test.
The intervals of monotonicity is tabulated in Table . . Interval −∞− , ) − ( , ∞ Sign of ′ f - - Monotonicity strictly decreasing strictly increasing strictly decreasing strictly increasing Table . By the first derivative test f x ( ) has local minimum at x = − , its local minimum value is − .
At x = , the function f x ( ) has local maximum at x = , and its local maximum value is . At x = , the function f x ( ) has local minimum at x = , and its local minimum value is − . Remark When the second derivative vanishes, we have no information about extrema. We have used the first derivative test to find out the extrema of the function!
Fig. . - - Example . Find the local maximum and minimum of the function x y on the line x = .
Let the given function be written as f x . Now, ( ) = x Therefore, ′ f ( ) = x x ′ f ( ) = x x , , ⇒ and ′′ ( ) = x The stationary numbers of f x ( ) are x = , , at these points the values of ′′ ( ) are respectively , − and . At x = , it has local minimum and its value is f ( ) . At x = , it has local maximum and