Locate any points of inflexion of the curve y = x³.

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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 possi-ble 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 of inflexion. A graph is shown in Figure 12.11. Note that at this point of inflexion y′ = 0 too.

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