Second Derivatives and Applications

Definition of the second derivative, concavity, inflexion points

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SL 5.7

The derivative of the derivative of a function is its second derivative:

f′′(x)=(f′(x))′🚫

dx2d2y=dxd(dxdy)=y′′🚫

SL 5.7

We determine concavity by the sign of f′′:

f′′{>0⇔f concave up<0⇔f concave down🚫

SL 5.7

At a stationary point (f′(a)=0),

Using the second derivative to classify a stationary point is often called the second derivative test.

SL 5.7

Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫

SL 5.7

SL 5.7

The derivative of the derivative of a function is its second derivative:

f′′(x)=(f′(x))′🚫

dx2d2y=dxd(dxdy)=y′′🚫

SL 5.7

We determine concavity by the sign of f′′:

f′′{>0⇔f concave up<0⇔f concave down🚫

SL 5.7

At a stationary point (f′(a)=0),

Using the second derivative to classify a stationary point is often called the second derivative test.

SL 5.7

Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫

SL 5.7