Math ↠ Curve sketching ↠ Inflection point

Inflection point

At an inflection point, the function graph changes its curvature behavior.

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Remember

Necessary criterion

The prerequisite for the presence of inflection points is that the second derivative has a zero at this point:
$f''(x_W)=0$

Sufficient criterion

An inflection point exists if in addition:
$f'''(x_W)\neq0$
Inflection point
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Method

  1. Find derivatives
  2. Calculate zero(s) of the second derivative
  3. Insert zero(s) in the third derivative
  4. Indicate inflection point(s)

Example

Find the inflection points of the function $f(x)=x^3+2x^2-4x-8$.

  1. Find derivatives

    $f'(x)=3x^2+4x-4$ (the first derivative is not needed)
    $f''(x)=6x+4$
    $f'''(x)=6$

  2. Calculate zeros of the second derivative

    $f''(x)=6x+4$
    $x_W\Leftrightarrow f''(x_W)=0$

    $6x+4=0\quad|-4$
    $6x=-4\quad|:6$
    $x_W=-\frac23$

  3. Insert zeros into the third derivative

    We use the places just identified in the third derivation.
    $f'''(x)=6$

    $f'''(-\frac23)=6\neq0$
    => There is an inflection point at this position $x=-\frac23$

    Note: We use the places just identified in the third derivation.

  4. Specify inflection points

    Inflection points should be specified: Therefore, calculate the y-coordinate with the original function.

    $f(-\frac23)$ $=(-\frac23)^3+2\cdot(-\frac23)^2-4\cdot(-\frac23)-8$ $=-4.74$
    => Inflection point: $W(-\frac23|-4.74)$