Question 12.5: Locate any points of inflexion of the curve y = x³....

Given y = x³ , then y’ = 3x² and y” = 6x. Points of inflexion can only occur where y” = 0 or does not exist. Clearly y” exists for all x and is zero when x = 0. It is possible that a point of inflexion occurs when x = 0 but we must examine the concavity of the curve on either side. To the left of x = 0, x is negative and so y” is negative. Hence to the left, the curve is concave down. To the right of x = 0, x is positive and so y” is positive. Hence to the right, the curve is concave up. Thus the concavity changes at x = 0. We conclude that x = 0 is a point ofinflexion. A graph is shown in Figure 12.11. Note that at this point of inflexion y’ = 0 too.