Increasing/Decreasing/Concavity

Definition

Any function ${\displaystyle f}$ is

• increasing at ${\displaystyle x}$ if ${\displaystyle f'(x)>0}$
• decreasing at ${\displaystyle x}$ if ${\displaystyle f'(x)<0}$
• concave up at ${\displaystyle x}$ if ${\displaystyle f''(x)>0}$
• concave down at ${\displaystyle x}$ if ${\displaystyle f''(x)<0}$

Critical point is the point where ${\displaystyle f'(x)=0}$. A critical point is

• maxima where ${\displaystyle f''(x)<0}$
• minima where ${\displaystyle f''(x)>0}$
• inflection point if ${\displaystyle f''(x)=0}$ and ${\displaystyle f''(x)}$ does change sign around ${\displaystyle x}$

Critical Points

Critical Points are points where ${\displaystyle f'(x)=0}$ Notice that at critical points the function is neither increasing not decreasing.

You can comment on increasing/decreasing on either side of critical points.