Increasing/Decreasing/Concavity

This article is part of the MathHelp Tutoring Wiki

Definition

Any function $f$ is

• increasing at $x$ if $f^{\prime }(x)>0$
• decreasing at $x$ if $f^{\prime }(x)<0$
• concave up at $x$ if $f^{″}(x)>0$
• concave down at $x$ if $f^{″}(x)<0$

Critical point is the point where $f^{\prime }(x)=0$. 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 $f^{\prime }(x)=0$ Notice that at critical points the function is neither increasing not decreasing.

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