Application of Derivatives • Topic 2 of 3

Increasing & Decreasing Functions

The sign of the first derivative tells you whether a function rises or falls.

The monotonicity test

On an interval $I$:

  • If $f'(x)>0$ for all $x\in I$, then $f$ is strictly increasing on $I$.
  • If $f'(x)<0$ for all $x\in I$, then $f$ is strictly decreasing on $I$.
  • If $f'(x)=0$ throughout, $f$ is constant.

Method

To find where $f$ increases or decreases: compute $f'(x)$, find the critical points where $f'(x)=0$ (or is undefined), and test the sign of $f'$ in each resulting interval. A sign chart organises this cleanly. The points where $f'$ changes sign separate increasing from decreasing behaviour.

Solved Examples
1
Worked Example
Find the intervals where $f(x)=x^2-4x+1$ is increasing/decreasing.
Solution

$f'(x)=2x-4=0\Rightarrow x=2$. For $x<2,\ f'<0$ (decreasing); for $x>2,\ f'>0$ (increasing). So $f$ decreases on $(-\infty,2)$ and increases on $(2,\infty)$.

2
Worked Example
Show $f(x)=x^3+x$ is increasing on all of $\mathbb{R}$.
Solution

$f'(x)=3x^2+1>0$ for every real $x$ (sum of a non-negative term and $1$). Hence $f$ is strictly increasing everywhere.

3
Worked Example
Find where $f(x)=x^3-3x$ is decreasing.
Solution

$f'(x)=3x^2-3=3(x-1)(x+1)$. This is negative for $-1

4
Worked Example
Is $f(x)=e^{x}$ increasing or decreasing?
Solution

$f'(x)=e^{x}>0$ for all $x$, so $e^{x}$ is strictly increasing on $\mathbb{R}$.

Key Points

  • $f'(x)>0 \Rightarrow$ increasing; $f'(x)<0 \Rightarrow$ decreasing on that interval.
  • Critical points: where $f'(x)=0$ or is undefined.
  • Build a sign chart of $f'$ to read off the intervals.
  • Sign changes of $f'$ separate increasing and decreasing stretches.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.$f$ is strictly increasing on $I$ if, for all $x\in I$:
Explanation: Positive first derivative means increasing.
Q2.$f(x)=x^2-4x+1$ is increasing on:
Explanation: $f'=2x-4>0$ when $x>2$.
Q3.$f(x)=x^3-3x$ is decreasing on:
Explanation: $f'=3(x-1)(x+1)<0$ for $-1
Q4.Which is increasing on all of $\mathbb{R}$?
Explanation: $\tfrac{d}{dx}e^x=e^x>0$ everywhere.
Practice more MCQs → 📝 Assignment · 35 marks