. . Extrema using Second Derivative Test The Second Derivative Test: The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum. Theorem .
(The Second Derivative Test) Suppose that c is a critical point at which ′ c ( ) , that ′ f ( ) exists in a neighborhood of c , and that f ( ) c exists. Then f has a relative maximum value at c if ′′ c ( ) and a relative minimum value at c if ′′ > c ( ) . If ′′ c ( ) , the test is not informative. .
- π - π π π Fig. . Table . - - Applications of Differential Calculus Example .
Find the local extremum of the function f x ( ) = . We have, ′ f ( ) = x + gives x = − ⇒ x = − and ′′ ( ) = x . As ′′ − > f ( , the function has local minimum at x =− . The local minimum value is f ( =− .
Therefore, the extreme point is ( . Example . Find the local extrema of the function f x ( ) = . Differentiating with respect to x, we get ′ ( ) x = - = x − = x ′ ( ) = Þ x =− , , .
Hence the critical numbers are x = − , , . Now, ′′ ( ) = Þ ′′ − ) = , ′′ ( ) = , ′′ ( ) = . As ′′ − and ′′ ( ) are positive by the second