Increasing Functions and Decreasing Functions

Increasing and Decreasing

We can determine intervals where a curve increases or decreases 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 increase or decrease for all $x \in \mathbb{R}$. These functions are called either monotone (monotonic) increasing or monotone (monotonic) decreasing.

$y=2^x$ is monotone (monotonic) increasing for all $x$
$y=2^{-x}$ is monotone (monotonic) decreasing for all $x$

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 increases when $f'(x) \gt 0$ and decreases when $f'(x) \lt 0$.

Example 1

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

\require{AMSsymbols} \require{color} \displaystyle \begin{align} f'(x) \gt 0 &&\color{red} \text{increasing} \\ 2x-4 \gt 0 \\ 2x \gt 4 \\ \therefore x \gt 2 \end{align}

Example 2

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

\require{AMSsymbols} \require{color} \displaystyle \begin{align} f^{\prime}(x) \lt 0 &&\color{red} \text{decreasing} \\ 6x^2-6x-12 \lt 0 \\ x^2-x-2 \lt 0 \\ (x+1)(x-2) \lt 0 \\ \therefore -1 \lt x \lt 2 \end{align}

Example 3

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

\require{AMSsymbols} \require{color} \displaystyle \begin{align} f'(x) \lt 0 &&\color{red} \text{decreasing} \\ 12x^3-24x^2 \lt 0 \\ x^3-2x^2 \lt 0 \\ x^2(x-2) \lt 0 \\ \therefore x \lt 0 \text{ and } x \gt 2 \end{align}

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