A Note A point c in the domain of a function f at which either f ′ ( c ) = or f is not differentiable is called a critical point of f . Note that if f is continuous at c and f ′ ( c ) = , then there exists an h > such that f is differentiable in the interval ( c – h , c + h ). We shall now give a working rule for finding points of local maxima or points of local minima using only the first order derivatives. Theorem (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 inflection (Fig . ). A Note If c is a point of local maxima of f , then f ( c ) is a local maximum value of f . Similarly, if c is a point of local minima of f ,