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Inflection point
An inflection point is a point on the curve where the curvature changes in nature. The shape of the curve there changes from concave to convex, or vice versa.
Example 1
Investigate the inflection points of the function f (x) = x4 − 6x3 + 12x2 − 8x + 1
f ′(x) = 4x3 − 18x2 + 24x − 8
f ′′(x) = 12x2 − 36x + 24
f ′′′(x) = 24x − 36
There are inflection points when f ′′(x) = 0 and f ′′′(x) ≠ 0
12x2 − 36x + 24 = 0 ⇒ x2 − 3x + 2 = 0 ⇒ x1 = 1 x2 = 2
f ′′′(1) = 24 − 36 = −12 ⇒ B1 (1, 0)
f ′′′(2) = 48 − 36 = 12 ⇒ B2 (2, 1)
The slope of an inflection point is m = f ′(xb)
B1 (1, 0) ⇒ m1 = f ′(1) = 4 − 18 + 24 − 8 = 2 increasing
B2 (2, 1) ⇒ m2 = f ′(2) = 32 − 72 + 48 − 8 = 0 horizontal