Math ↠ Curve sketching ↠ Monotony behavior

Monotony behavior

Monotony behavior is the slope behavior of a function. This tells you when a function is going up or down.

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General

A function is for two places $x_1$ and $x_2$ with $x_1<x_2$

  • increasing monotonically if $f(x_1)\le f(x_2)$
  • decreasing monotonically if $f(x_1)\ge f(x_2)$
  • strictly (monotonically) increasing if $f(x_1)< f(x_2)$
  • strictly (monotonically) decreasing if $f(x_1)> f(x_2)$
Monotony behavior
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Remember

You can also determine the monotony behavior in an interval with the derivation.

The function is differentiable on the interval $I$ and

  • increasing monotonically if for all $x\in I$ we have $f'(x)\ge0$
  • decreasing monotonically if for all $x\in I$ we have $f'(x)\le0$
  • strictly monotonically increasing if for all $x\in I$ we have $f'(x)>0$
  • strictly monotonically decreasing if for all $x\in I$ we have $f'(x)<0$
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Method

  1. Find derivative
  2. Calculate the zero (s) of the derivative
  3. Determine intervals
  4. Sample insertion into the derivative

Example

Investigate the function $f(x)=x^3+2x^2-4x-8$ for monotory.

  1. Find derivative

    $f'(x)=3x^2+4x-4$

  2. Calculate the zero of the derivative

    There is a quadratic equation that can be solved, for example, with the PQ formula.
    $f'(x)=3x^2+4x-4\quad|:3$
    $f'(x)=x^2+\frac43x-\frac43$

    $x_{1.2} = -\frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$
    $x_{1.2} = -\frac{2}{3} \pm\sqrt{(\frac23)^2+\frac43}$
    $x_{1.2} = -\frac{2}{3} \pm\sqrt{\frac{16}{9}}$
    $x_{1.2} = -\frac{2}{3} \pm\frac43$
    $x_1=\color{blue}{-2} \quad x_2=\color{green}{\frac23}$

  3. Determine intervals

    With the zeros of the derivative function you form the intervals at which the monotony behavior is investigated.
    $x_1=\color{blue}{-2} \quad x_2=\color{green}{\frac23}$

    $I_1(-\infty|\color{blue}{-2})$, $I_2(\color{blue}{-2}|\color{green}{\frac23})$, $I_3(\color{green}{\frac23}|\infty)$

  4. Sample inserts in the derivative

    Use any value from each interval in the derivative.

    $I_1(-\infty|-2)$:
    Sample insertion: $x=\color{red}{-3}$
    $f'(\color{red}{-3})=3\cdot(\color{red}{-3})^2+4\cdot(\color{red}{-3})-4$ $=11 > 0$
    => The derivative is positive in interval $I_1$, i.e. the function is monotonically increasing in this interval.

    $I_2(-2|\frac23)$:
    Sample insertion: $x=\color{red}{0}$
    $f'(\color{red}{0})=3\cdot\color{red}{0}^2+4\cdot\color{red}{0}-4$ $=-4 < 0$
    => The derivative is negative in the interval $I_2$, i.e. the function is monotonically decreasing in this interval.

    $I_3(\frac23|\infty)$:
    Sample insertion: $x=\color{red}{1}$
    $f'(\color{red}{1})=3\cdot\color{red}{1}^2+4\cdot\color{red}{1}-4$ $=3 > 0$
    => The derivative is positive in the interval $I_3$, i.e. the function is monotonically increasing in this interval.