Summary of Derivatives and the Shape of a Graph

Essential Concepts

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

Glossary

concave down
if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is decreasing over [latex]I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex]
concave up
if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is increasing over [latex]I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex]
concavity
the upward or downward curve of the graph of a function
concavity test
suppose [latex]f[/latex] is twice differentiable over an interval [latex]I[/latex]; if [latex]f^{\prime \prime}>0[/latex] over [latex]I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex]; if [latex]f^{\prime \prime}<0[/latex] over [latex]I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex]
first derivative test
let [latex]f[/latex] be a continuous function over an interval [latex]I[/latex] containing a critical point [latex]c[/latex] such that [latex]f[/latex] is differentiable over [latex]I[/latex] except possibly at [latex]c[/latex]; if [latex]f^{\prime}[/latex] changes sign from positive to negative as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] has a local maximum at [latex]c[/latex]; if [latex]f^{\prime}[/latex] changes sign from negative to positive as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] has a local minimum at [latex]c[/latex]; if [latex]f^{\prime}[/latex] does not change sign as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] does not have a local extremum at [latex]c[/latex]
inflection point
if [latex]f[/latex] is continuous at [latex]c[/latex] and [latex]f[/latex] changes concavity at [latex]c[/latex], the point [latex](c,f(c))[/latex] is an inflection point of [latex]f[/latex]
second derivative test
suppose [latex]f^{\prime}(c)=0[/latex] and [latex]f^{\prime \prime}[/latex] is continuous over an interval containing [latex]c[/latex]; if [latex]f^{\prime \prime}(c)>0[/latex], then [latex]f[/latex] has a local minimum at [latex]c[/latex]; if [latex]f^{\prime \prime}(c)<0[/latex], then [latex]f[/latex] has a local maximum at [latex]c[/latex]; if [latex]f^{\prime \prime}(c)=0[/latex], then the test is inconclusive