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:$f'''(x_W)\neq0$
i
Method
- Find derivatives
- Calculate zero(s) of the second derivative
- Insert zero(s) in the third derivative
- Indicate inflection point(s)
Example
Find the inflection points of the function $f(x)=x^3+2x^2-4x-8$.
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Find derivatives
$f'(x)=3x^2+4x-4$ (the first derivative is not needed)
$f''(x)=6x+4$
$f'''(x)=6$ -
Calculate zeros of the second derivative
$f''(x)=6x+4$
$x_W\Leftrightarrow f''(x_W)=0$
$6x+4=0\quad|-4$
$6x=-4\quad|:6$
$x_W=-\frac23$ -
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. -
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: $W(-\frac23|-4.74)$
Learning
- Curve sketching
- Intersection and Zero of a function
- Symmetry behavior
- Monotony behavior
- Curvature behavior
- Maxima and minima (Extrema)
- Inflection point
Exercises
Premium
- Grundlagen
- Symmetrieverhalten
- Monotonieverhalten
- Extrem- und Wendepunkte