) be a critical point of function f x ( ) that is continuous on an open interval I containing c . If f x ( ) is differentiable on the interval, except possibly at c , then f c ( ) can be classified as follows:(when moving across I from left to right) (i) If ′ ( ) x changes from negative to positive at c , then f x ( ) has a local minimum f ( c ). (ii) If ′ ( ) x changes from positive to negative at c , then f x ( ) has a local maximum f ( c ). (iii) If ′ ( ) x is positive on both sides of c , or negative on both sides of c then f x ( ) has neither a local minimum nor a local minimum.
• Second Derivative Test Suppose that c is a critical point at which ′ ( ) = , that ′′ ( ) x exists in a neighbourhood of c , and that ′ ( ) c exists. Then f has a relative maximum value at c if ′′ ( ) < and a relative minimum value at c if ′′ ( ) > . If ′′ ( ) = , the test is not informative.