Increasing/Decreasing/Concavity

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Definition

Any function $f$ is

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

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

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

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Critical Points

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

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