Summary of Derivatives and the Shape of a Graph

If $c$ is a critical point of $f$ and $f^{\prime}(x)>0$ for [latex]xc[/latex], then $f$ has a local maximum at $c$ .
If $c$ is a critical point of $f$ and $f^{\prime}(x)<0$ for [latex]x0[/latex] for $x>c$ , then $f$ has a local minimum at $c$ .
If $f^{\prime \prime}(x)>0$ over an interval $I$ , then $f$ is concave up over $I$ .
If $f^{\prime \prime}(x)<0$ over an interval $I$ , then $f$ is concave down over $I$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$ , then $f$ has a local minimum at $c$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)<0$ , then $f$ has a local maximum at $c$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)=0$ , then evaluate $f^{\prime}(x)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c$ , to determine whether $f$ has a local extremum at $c$ .
If $f^{\prime \prime}(x)>0$ over an interval $I$ , then $f$ is concave up over $I$ .
If $f^{\prime \prime}(x)<0$ over an interval $I$ , then $f$ is concave down over $I$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$ , then $f$ has a local minimum at $c$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)<0$ , then $f$ has a local maximum at $c$ .
If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)=0$ , then evaluate $f^{\prime}(x)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c$ , to determine whether $f$ has a local extremum at $c$ .

Glossary

concave down: if $f$ is differentiable over an interval $I$ and $f^{\prime}$ is decreasing over $I$ , then $f$ is concave down over $I$

concave up: if $f$ is differentiable over an interval $I$ and $f^{\prime}$ is increasing over $I$ , then $f$ is concave up over $I$

concavity test: suppose $f$ is twice differentiable over an interval $I$ ; if $f^{\prime \prime}>0$ over $I$ , then $f$ is concave up over $I$ ; if $f^{\prime \prime}<0$ over $I$ , then $f$ is concave down over $I$

first derivative test: let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c$ ; if $f^{\prime}$ changes sign from positive to negative as $x$ increases through $c$ , then $f$ has a local maximum at $c$ ; if $f^{\prime}$ changes sign from negative to positive as $x$ increases through $c$ , then $f$ has a local minimum at $c$ ; if $f^{\prime}$ does not change sign as $x$ increases through $c$ , then $f$ does not have a local extremum at $c$

inflection point: if $f$ is continuous at $c$ and $f$ changes concavity at $c$ , the point $(c,f(c))$ is an inflection point of $f$

second derivative test: suppose $f^{\prime}(c)=0$ and $f^{\prime \prime}$ is continuous over an interval containing $c$ ; if $f^{\prime \prime}(c)>0$ , then $f$ has a local minimum at $c$ ; if $f^{\prime \prime}(c)<0$ , then $f$ has a local maximum at $c$ ; if $f^{\prime \prime}(c)=0$ , then the test is inconclusive