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Second Derivatives and Applications
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Definition of the second derivative, concavity, inflexion points
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Second Derivative
AHL AI 5.10
The derivative of the derivative of a function is its second derivative:
f′′(x)=(f′(x))′🚫
dx2d2y=dxd(dxdy)=y′′🚫
Concavity
AHL AI 5.10
We determine concavity by the sign of f′′:
f′′{>0⇔f concave up<0⇔f concave down🚫
Classifying stationary points using the second derivative
AHL AI 5.10
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.
Inflexion Points
AHL AI 5.10
Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫
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