Applications of Derivatives

Increasing and Decreasing Intervals

Find where f(x)=x³-3x increases and decreases using sign analysis of f'(x).

increasing-decreasing A function increases on an interval where f'(x)>0 and decreases where f'(x)<0. Testing a sample point in each interval between critical points determines the sign of the derivative on that interval.

Example

Use derivative signs on intervals to decide increasing or decreasing behavior.

highlighted = computed this step

Step 1 — Set up

Use f prime for the sign test.

f′(x)=3x2−3f'(x)= 3x^{2}-3

Step 2 — Critical points

Split the number line at -1 and 1.

x=-1x=1x= \hl{-1} \quad x= \hl{1}

Step 3 — Test interval 1

Test the sign of f prime on this interval.

(−∞,−1): f′>0 ? positive⇒increasing(-\infty, -1 ) :\ f'> 0 \ ?\ \hl{positive} \Rightarrow \hl{increasing}

Step 4 — Test interval 2

Test the sign of f prime on this interval.

(−1,1): f′<0 ? negative⇒decreasing( -1 , 1 ) :\ f'< 0 \ ?\ \hl{negative} \Rightarrow \hl{decreasing}

Step 5 — Test interval 3

Test the sign of f prime on this interval.

(1,∞): f′>0 ? positive⇒increasing( 1 ,\infty) :\ f'> 0 \ ?\ \hl{positive} \Rightarrow \hl{increasing}