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# Patterns of Inflection

A point of inflection of a curve $y=f(x)$ is a point at which the second derivative $\frac{d^2y}{dx^2}$ changes sign.

Geometrically, you can think of a point of inflection as a point where the tangent to the curve crosses the curve.

Points of inflection need not also be stationary points (first derivative also zero), although they might be.

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A point of inflection of a curve $y=f(x)$ is a point at which the second derivative $\frac{d^2y}{dx^2}$ changes sign.

Geometrically, you can think of a point of inflection as a point where the tangent to the curve crosses the curve.

Points of inflection need not also be stationary points (first derivative also zero), although they might be.

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.