Find any maximum points, minimum points and points of inflexion of y = x³ + 2x².

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Given y = x³ + 2x² then y′ = 3x² + 4x and y′′ = 6x + 4. Let us first find any maximum and minimum points. The first derivative y′ is zero when 3x² + 4x = x(3x + 4) = 0, that is when x = 0 or x=-\frac{4}{3}. Using the second-derivative test we nd y′′(0) = 4 which corresponds to a minimum point. Similarly, y^{\prime \prime}\left(-\frac{4}{3}\right)=-4 which corresponds to a maximum point.

We seek points of inflexion by looking for points where y′′ = 0 and then examining the concavity on either side. y′′ = 0 when x=-\frac{2}{3}.

Since y′′ is negative when x<-\frac{2}{3}, then y′ is decreasing there, that is the function is concave down. Also, y′′ is positive when x>-\frac{2}{3} and so y′ is then increasing, that is the function is concave up. Hence there is a point of inflexion when x=-\frac{2}{3}. The graph of y = x³ + 2x² is shown in Figure 12.13.

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