Increasing & Decreasing Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Increasing & decreasing functions

What are increasing and decreasing functions?

• A function,$f\left(x\right)$, is increasing if${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{>}\mathbf{0}$

  • This means the value of the function (‘output’) increases as$x$ increases

• A function,$f\left(x\right)$, is decreasing if${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{<}\mathbf{0}$

  • This means the value of the function (‘output’) decreases as$x$ increases

• A function,$f\left(x\right)$, is stationary where${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{0}$

How do I find where functions are increasing, decreasing or stationary?

• To identify the intervals on which a function is increasing or decreasing 

STEP 1

Find the derivative ${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

STEP 2

Solve the inequalities

${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{>}\mathbf{0}$ (for increasing intervals) and/or

${\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{<}\mathbf{0}$ (for decreasing intervals)

• Most functions are a combination of increasing intervals, decreasing intervals and stationary points

  • a range of values of$x$ (interval) is given where a function satisfies each condition

  • e.g.  The function$f\left(x\right)=x^2$ has derivative$f^{\text{'}}\left(x\right)=2x$ so

    • $f\left(x\right)$ is decreasing for $x<0$

    • $f\left(x\right)$ is stationary at $x=0$

    • $f\left(x\right)$ is increasing for $x>0$