point of inflexion derivative rule

Point of inflexion

The function ƒ is increasing before x =0, and also it is increasing after x=0 such a point of the function is called the point of inflexion. Point of inflexion derivative rule detail:

Point of inflexion


(1) A stationary point is called turning if it either a maximum point or a minimum point.

(2) if ƒ’(x) >0 before the point x =a ,ƒ’(x) = 0 before the point x =a and ƒ’(x) >0 after x =a

then ƒ does not have a relative maxima.

See the graph of ƒ(x) = x³ in this case we have

ƒ’(x) =3x²

ƒ’(x)(0- ε) = 3(-ε)²= 3ε²>0


ƒ’(x)(0+ ε) = 3(ε)²= 3ε²>0

which point has these two conditions is called the point of inflexion.

Point of inflexion derivative rule:

What is the first Derivative rule:

Let ƒ be a differentiable in neighborhood where ƒ’(c)=0

Relative maxima of ƒ:

(1) if ƒ’(x) changes sign positive to negative as x increases through c, then ƒ(c)is the relative maxima of ƒ.

Relative minima of ƒ:

(2) if ƒ’(x) changes sign from negative to positive as x increases through c, then if ƒ’(c) is called the relative minima of ƒ.