Increasing Functions and Decreasing Functions

Increasing and Decreasing

We can determine intervals where a curve is increasing or decreasing by considering $f'(x)$ on the interval in question.
  • $f'(x) \gt 0$: $f(x)$ is increasing
  • $f'(x) \lt 0$: $f(x)$ is decreasing

Monotone (Monotonic) Increasing or Decreasing

Many functions are either increasing or decreasing for all $x \in \mathbb{R}$. These functions are called as either monotone (monotonic) increasing or monotone (monotonic) decreasing.
$y=2^x$ is monotone (monotonic) increasing for all $x$
Monotone Increasing Function
$y=2^{-x}$ is monotone (monotonic) decreasing for all $x$
Monotone Decreasing Function Geek Nerd Note That 2 Note:
Ensure that $f'(x)=0$ indicates the curve $y=f(x)$ is stationary, so the curve is neither increasing nor decreasing when $f'(x) = 0$. This means that the curve is increasing when $f'(x) \gt 0$ and the curve is decreasing when $f'(x) \lt 0$.

Example 1

Find intervals where $f(x)=x^2-4x+3$ is increasing.

Example 2

Find intervals where $f(x)=2x^3-3x^2-12x+5$ is decreasing.

Example 3

Find intervals where $f(x)=3x^4-8x^3+2$ is decreasing.