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Second Derivatives and Applications
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Second derivatives \(f''\), using the sign of \(f''\) to determine concavity, locate inflexion points where \(f''(x)=0\) and changes sign, and classify stationary points with the second derivative test.
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Second Derivative
SL 5.7
The derivative of the derivative of a function is its second derivative:
f′′(x)=(f′(x))′🚫
dx2d2y=dxd(dxdy)=y′′🚫
Concavity
SL 5.7
We determine concavity by the sign of f′′:
f′′{>0⇔f concave up<0⇔f concave down🚫
Classifying stationary points using the second derivative
SL 5.7
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
SL 5.7
Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫
Graphs of f, f' and f''
SL 5.7
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.
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