. . Extrema using First Derivative Test After we have determined the intervals on which a function is increasing or decreasing, it is not difficult to locate the relative extrema of the function. The location or points at which the relative extrema occurs for a given function f x ( ) can be observed through the graph y ( ) .
However to find the exact point and the value of the extrema of functions we need to use certain test on functions. One such test is the first derivative test, which is stated in the following theorem. Theorem . (First Derivative Test) Let ( , ( )) c f c be a critical point of function f x ( ) that is continuous on an open interval I containing c .
If ( ) is differentiable on the interval, except possibly at c , then f c ( ) can be classified as follows: (when moving across the interval I from left to right) (i) If ′ f ( ) changes from negative to positive at c , then f x ( ) has a local minimum f c ( ) . (ii) If ′ f ( ) changes from positive to negative at c , then f x ( ) has a local maximum f c ( ) . (iii) If ′ f ( ) is positive on both sides of c or negative on both sides of c , then f c ( ) is neither a local minimum nor a local maximum. Fig.
. c c c ( , ( )) c f c ( , ( )) f c ( , ( )) f c ′ f c ( ) ′ f c ( ) ′ f c ( ) does not exist f c ( ) is a local maximum f c ( ) is not a local extremum f c ( ) is a local minimum ++++++++++ ++++++++++ –––––– –––––– ′ > x ( ) ′ > x ( ) ′ x ( ) ′ x ( )