Sign of ′′ ( ) _ Concavity concave up concave down concave up The curve is concave upwards on ( , ) −∞ and ( , ¥ . The curve is concave downwards on ( , ) . As ′′ ( ) changes its sign when it passes through x = and x , ( , ( )) ( , ) and ( , ( )) ( , are points of inflection for the graph y ( ) . The sign change may be observed from the adjoining figure of the curve ′′ ( ) .
Example . Determine the intervals of concavity of the curve y The given function is a periodic function with period p and hence there will be stationary points and points of inflections in each period interval. We have, dx = cos x and d y = − sin Now, d y = − ⇒ sin x π . We now consider an interval, ( , ) −π π by splitting into two sub intervals ( , ) −π and ( , ) p .
In the interval ( , ) −π , d y > and hence the function is concave upward. In the interval ( , ), d y and hence the function is concave downward. Therefore ( , ) is a point of inflection (see Fig. .
). The general intervals need to be considered to discuss the concavity of the curve are ( ,( ) ) + , where n is any integer which can be discussed as before to conclude that ( , ) n p are also points of inflection.