Applications of Derivatives

Concavity and Inflection Points

Find concavity of f(x)=x³-3x using f''(x)=6x and locate the inflection at x=0.

Example

Use second-derivative signs to decide concavity and locate inflection.

highlighted = computed this step

Step 1 — Set up

Use f double prime for concavity.

f′′(x)=6xf''(x)= 6x

Step 2 — Candidate inflection

Set f double prime equal to 0.

6x=0⇒x=06x = 0 \Rightarrow x= \hl{0}

Step 3 — Test concavity 1

Test the sign of f double prime on this interval.

(−∞,0): f′′<0 ? negative⇒concave down(-\infty, 0 ) :\ f''< 0 \ ?\ \hl{negative} \Rightarrow \hl{concave down}

Step 4 — Test concavity 2

Test the sign of f double prime on this interval.

(0,∞): f′′>0 ? positive⇒concave up( 0 ,\infty) :\ f''> 0 \ ?\ \hl{positive} \Rightarrow \hl{concave up}

Step 5 — Inflection point

The inflection point is at the origin.

(0,0)( \hl{0} , \hl{0} )
concavity-and-inflection A function is concave up where f''(x)>0 and concave down where f''(x)<0. An inflection point occurs where f''(x)=0 and concavity changes. Setting f''(x)=0 and checking sign changes identifies inflection points.