Derivatives: the basics

Local maxima and minima

A local maximum or minimum occurs where f′(x)=0f'(x)=0 and the sign of f′f' changes.

f′(a)=0, f′: +→− ⇒ local maxf′(a)=0, f′: −→+ ⇒ local min\begin{gathered} f'(a)=0,\ f':\ +\to -\ \Rightarrow\ \text{local max} \\[4pt] f'(a)=0,\ f':\ -\to +\ \Rightarrow\ \text{local min} \end{gathered}

Points where f′(x)=0f'(x)=0 are the candidates. If the sign of f′f' changes there, you have an extremum.

f′: +→− local max,f′: −→+ local min\begin{gathered} f':\ +\to -\ \text{local max}, \\[4pt] f':\ -\to +\ \text{local min} \end{gathered}

If the sign does not change, it is not an extremum. Note that f′(x)=0f'(x)=0 alone does not decide it.

xylocal maxlocal minincreasingdecreasing
A local maximum where f' changes from + to −, a local minimum where it changes from − to +.Drag the dot

Examples

  1. 01If this is f'(x) for a cubic f(x), at which x does f have a local minimum?f′(x)=3(x+3)(x+1)\displaystyle f'(x)=3(x+3)(x+1)AnswerHide
    Answer−1\displaystyle -1

    Steps

    1. x<−3\displaystyle x<-3⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0−3<x<−1\displaystyle -3<x<-1⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>−1\displaystyle x>-1⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0
  2. 02If this is f'(x) for a cubic f(x), at which x does f have a local maximum?f′(x)=3(x−3)(x−4)\displaystyle f'(x)=3(x-3)(x-4)AnswerHide
    Answer3\displaystyle 3

    Steps

    1. x<3\displaystyle x<3⇒f′(x)>0\displaystyle \Rightarrow f'(x)>03<x<4\displaystyle 3<x<4⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>4\displaystyle x>4⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0
  3. 03If this is f'(x) for a cubic f(x), at which x does f have a local maximum?f′(x)=3(x+4)(x−1)\displaystyle f'(x)=3(x+4)(x-1)AnswerHide
    Answer−4\displaystyle -4

    Steps

    1. x<−4\displaystyle x<-4⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0−4<x<1\displaystyle -4<x<1⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>1\displaystyle x>1⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0
  4. 04If this is f'(x) for a cubic f(x), at which x does f have a local maximum?f′(x)=3(x+1)(x−2)\displaystyle f'(x)=3(x+1)(x-2)AnswerHide
    Answer−1\displaystyle -1

    Steps

    1. x<−1\displaystyle x<-1⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0−1<x<2\displaystyle -1<x<2⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>2\displaystyle x>2⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0
  5. 05If this is f'(x) for a cubic f(x), at which x does f have a local maximum?f′(x)=3(x+4)(x−4)\displaystyle f'(x)=3(x+4)(x-4)AnswerHide
    Answer−4\displaystyle -4

    Steps

    1. x<−4\displaystyle x<-4⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0−4<x<4\displaystyle -4<x<4⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>4\displaystyle x>4⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0
  6. 06If this is f'(x) for a cubic f(x), at which x does f have a local maximum?f′(x)=3(x+2)(x+1)\displaystyle f'(x)=3(x+2)(x+1)AnswerHide
    Answer−2\displaystyle -2

    Steps

    1. x<−2\displaystyle x<-2⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0−2<x<−1\displaystyle -2<x<-1⇒f′(x)<0\displaystyle \Rightarrow f'(x)<0x>−1\displaystyle x>-1⇒f′(x)>0\displaystyle \Rightarrow f'(x)>0

FAQ

What is the difference between a local maximum and the maximum?
A local maximum is the largest value nearby. The maximum is the largest value on the whole interval, and you decide it by also comparing the values at the endpoints.

Published 2026-09-03 · Updated 2026-09-05