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Inflection Point

See how to find the inflection point.
1 examples and their solutions.

Concave Up, Concave Down

Concave Up


f'(x) + -
f''(x) + +
f(x)
If f''(x) is (+),
then the slope of y = f(x), f'(x), increases.
Left graph
f'(x): 1, 2, 3, ...
Right graph
f'(x): -3, -2, -1, ...
Then y = f(x) is concave up.

Concave Down


f'(x) + -
f''(x) - -
f(x)
If f''(x) is (-),
then the slope of y = f(x), f'(x), decreases.
Left graph
f'(x): 3, 2, 1, ...
Right graph
f'(x): -1, -2, -3, ...
Then y = f(x) is concave down.

Inflection Point

Definition

An inflection point is the point
where the sign of f''(x) changes.
Below are the examples of the inflection points.

f'(x) +
f''(x) + → 0 → -
f(x)

f'(x) + → 0 → +
f''(x) + → 0 → -
f(x)

f'(x) -
f''(x) - → 0 → +
f(x)

f'(x) - → 0 → -
f''(x) - → 0 → +
f(x)

Example

f(x) = x3 - 3x2 + 6

1. Local maximum, minimum?
2. Inflection point?
Solution