Inflection Points
At an inflection point, the function is not concave or convex but is changing from concavity to convexity or vice versa.
If f and f' are differentiable at a.
Calculation of the Points of Inflection
Calculate the inflection points of:
f(x) = x3 − 3x + 2
To find the inflection points, follow these steps:
1. Find the second derivative and calculate its roots.
f''(x) = 6x 6x = 0 x = 0.
2. Determine the 3rd derivative and calculate the sign that the zeros take from the second derivative and if:
f'''(x) ≠ 0 There is an inflection point.
f'''(x) = 6 It is an inflection point.
3. Calculate the image (in the function) of the point of inflection.
f(0) = (0)3 − 3(0) + 2 = 2
Inflection point: (0, 2)
Example
Calculate the points of inflection of the function:
Domain:
There is an inflection point at x = 0, since the function changes from concave to convex.
Inflection point (0, 0)
