Sign of the derivative and where a function increases or decreases, with practice problems

Not necessarily. For y=x^3, f'(0)=0, but the function increases on both sides. Check whether the sign actually changes.

\begin{gathered} f'(x)>0 \Rightarrow \text{increasing}, \\[4pt] f'(x)<0 \Rightarrow \text{decreasing} \end{gathered}

The derivative is the slope at each point. The function increases where $f'(x)>0$ and decreases where $f'(x)<0$ .

A sign chart checks whether the sign changes at the points where $f'(x)=0$ .

f(x)=x³−3x. Orange where f'>0 (increasing), blue where f'<0 (decreasing). The boundaries are where f'=0.Drag the dot

Examples

01For which x is f'(x) positive? $\displaystyle f'(x)=3x-3$ AnswerHide

Answer $\displaystyle x>1$
02If this is f'(x), on which interval is f(x) decreasing? $\displaystyle f'(x)=6x^2-24$ AnswerHide

Answer $\displaystyle -2<x<2$
03If this is f'(x), on which interval is f(x) increasing? $\displaystyle f'(x)=4x-16$ AnswerHide

Answer $\displaystyle x>4$
04f'(x) is positive up to the boundary □. What is □? $\displaystyle f'(x)=2x-10 \;\Rightarrow\; x>\square$ AnswerHide

Answer $\displaystyle 5$
05If this is f'(x), on which interval is f(x) decreasing? $\displaystyle f'(x)=3x^2-48$ AnswerHide

Answer $\displaystyle -4<x<4$
06For which x is f'(x) positive? $\displaystyle f'(x)=4x-4$ AnswerHide

Answer $\displaystyle x>1$

Q1Does the function always change direction where

f'(x)=0

?Show answerHide

Not necessarily. For

y=x^3

f'(0)=0

, but the function increases on both sides. Check whether the sign actually changes.

Practice this pattern in the app, 5 problems a day.

Before this

The derivative function f'(x)coming soonGraphs as pictures of equations

This topic

Sign of f' and increasing/decreasing

Published 2026-09-03 · Updated 2026-09-05