Monotonicity condition extremum | Wiskunde op Tilburg University

Introduction: A point

$x$ such that

$y'(x)=0$ is called a stationary point of the function

$y(x)$ .

Theorem: Assume

$c$ is a stationary point of the function

$y(x)$ . It holds that:

if $y'(x)< 0$ for each $x<c$ in the neighborhood of $c$ and $y'(x)> 0$ for each $x>c$ in the neighborhood of $c$ , then $y(c)$ is a minimum
if $y'(x)> 0$ for each $x< c$ in the neighborhood of $c$ and $y'(x)< 0$ for each $x>c$ in the neighborhood of $c$ , then $y(c)$ is a maximum.