Definition of the second derivative, concavity, inflexion points
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Practice exam-style second derivatives and applications problems
The derivative of the derivative of a function is its second derivative:
We determine concavity by the sign of f′′:
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At a stationary point (f′(a)=0),
If f′′(a)>0, then f has a local minimum at x=a.
If f′′(a)<0, then f has a local maximum at x=a.
Using the second derivative to classify a stationary point is often called the second derivative test.
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Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫
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When f′ crosses the x-axis f has a maximum (f′′<0) or minimum (f′′>0)
When f′′ crosses the x-axis, f has an inflexion point.