Graphs and properties of power functions

Properties of

y=x^n

for natural

n

On $(0,+\infty)$ the function $y=x^n$ is strictly increasing for any $n\in\mathbb{N}$ . On $(-\infty,0)$ for even $n$ it decreases (“bowl” symmetry); for odd $n$ it increases again (branches through the origin). The point $(0;0)$ for $n>1$ is special: flatter touch of the $x$ -axis for larger even $n$ ; the familiar inflection of the cubic.

Global minimum of

y=x^n

for even

n

is at

x=0

; for odd

n>1

no extremum on the whole axis

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