® First Derivative Test Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then (i) If f ′ ( x ) changes sign from positive to negative as x increases through c, i.e., if f ′ ( x ) > at every point sufficiently close to and to the left of c , and f ′ ( x ) < at every point sufficiently close to and to the right of c , then c is a point of local maxima . (ii) If f ′ ( x ) changes sign from negative to positive as x increases through c , i.e., if f ′ ( x ) < at every point sufficiently close to and to the left of c , and f ′ ( x ) > at every point sufficiently close to and to the right of c , then c is a point of local minima .
(iii) If f ′ ( x ) does not change sign as x increases through c , then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion .