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
  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
  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
  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
  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
  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

FAQ

Q1What is the difference between a local maximum and the maximum?Show answerHide
A
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.

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