All courses › Calculus › Extreme points

Extreme points

At an extreme point (a maximum or minimum) the slope of the tangent is zero. The sign of the second derivative tells us whether it is a maximum or a minimum.

f'(x) = 0

critical point

f''(x) > 0 \Rightarrow \text{minimum},\quad f''(x) < 0 \Rightarrow \text{maximum}

the second derivative test

Symbols

$f'(x)$	slope of the tangent
$f''(x)$	curvature (concave up/down)

Example

$f(x)=x^2-4x$ : $f'(x)=2x-4=0 \Rightarrow x=2$ , and $f''(x)=2>0$ , so $x=2$ is a minimum, $f(2)=-4$ .

f'(x)=0

alone does not guarantee an extreme point - also check

f''(x)

, or whether the sign of

f'

changes.

Practise differentiation for free →

← Standard derivatives · Definite integral →

Part of Calculus: Differentiation.