Home/Differential Calculus/Function analysis/Critical points and extremaSection 8.3Critical points and extremaFormulaCopy LaTeXf′(c)=0 or f′(c) does not exist⇒c is a critical pointf'(c)=0\ \text{or}\ f'(c)\text{ does not exist} \Rightarrow c\text{ is a critical point}f′(c)=0 or f′(c) does not exist⇒c is a critical pointccc belongs to the domain of fff and to the interior of the interval under consideration.View detail →FormulaCopy LaTeXf′(c)=0, f′′(c)>0⇒f has a local minimum at cf'(c)=0,\ f''(c)>0 \Rightarrow f\text{ has a local minimum at }cf′(c)=0, f′′(c)>0⇒f has a local minimum at cSecond derivative test (minimum).View detail →FormulaCopy LaTeXf′(c)=0, f′′(c)<0⇒f has a local maximum at cf'(c)=0,\ f''(c)<0 \Rightarrow f\text{ has a local maximum at }cf′(c)=0, f′′(c)<0⇒f has a local maximum at cSecond derivative test (maximum).View detail →FormulaCopy LaTeXf′(x) changes from + to − on c⇒f local maximum at cf'(x)\text{ changes from }+\text{ to }-\text{ on }c \Rightarrow f\text{ local maximum at }cf′(x) changes from + to − on c⇒f local maximum at cFirst derivative test.View detail →FormulaCopy LaTeXf′(x) changes from − to + on c⇒f local minimum at cf'(x)\text{ changes from }-\text{ to }+\text{ on }c \Rightarrow f\text{ local minimum at }cf′(x) changes from − to + on c⇒f local minimum at cView detail →
FormulaCopy LaTeXf′(c)=0 or f′(c) does not exist⇒c is a critical pointf'(c)=0\ \text{or}\ f'(c)\text{ does not exist} \Rightarrow c\text{ is a critical point}f′(c)=0 or f′(c) does not exist⇒c is a critical pointccc belongs to the domain of fff and to the interior of the interval under consideration.View detail →
FormulaCopy LaTeXf′(c)=0, f′′(c)>0⇒f has a local minimum at cf'(c)=0,\ f''(c)>0 \Rightarrow f\text{ has a local minimum at }cf′(c)=0, f′′(c)>0⇒f has a local minimum at cSecond derivative test (minimum).View detail →
FormulaCopy LaTeXf′(c)=0, f′′(c)<0⇒f has a local maximum at cf'(c)=0,\ f''(c)<0 \Rightarrow f\text{ has a local maximum at }cf′(c)=0, f′′(c)<0⇒f has a local maximum at cSecond derivative test (maximum).View detail →
FormulaCopy LaTeXf′(x) changes from + to − on c⇒f local maximum at cf'(x)\text{ changes from }+\text{ to }-\text{ on }c \Rightarrow f\text{ local maximum at }cf′(x) changes from + to − on c⇒f local maximum at cFirst derivative test.View detail →
FormulaCopy LaTeXf′(x) changes from − to + on c⇒f local minimum at cf'(x)\text{ changes from }-\text{ to }+\text{ on }c \Rightarrow f\text{ local minimum at }cf′(x) changes from − to + on c⇒f local minimum at cView detail →