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) | ⤷ | ⤷ |
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) | ⤶ | ⤶ |
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.
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 1. Local maximum, minimum?
2. Inflection point?
f(x) = x3 - 3x2 + 6
f'(x) = 3x2 - 3⋅2x1 + 0 - [1] [2]
= 3x2 - 6x
= 3x(x - 2) = 0 - [3]
x = 0, 2
f''(x) = 3⋅2x1 - 6 - [5]
= 6x - 6
= 6(x - 1) = 0
x = 1
[7]
f'(x) = 3x2 - 3⋅2x1 + 0 - [1] [2]
= 3x2 - 6x
= 3x(x - 2) = 0 - [3]
x = 0, 2
f''(x) = 3⋅2x1 - 6 - [5]
= 6x - 6
= 6(x - 1) = 0
x = 1
| x | ... | 0 | ... | 1 | ... | 2 | ... |
|---|---|---|---|---|---|---|---|
| f'(x) | |||||||
| f''(x) | |||||||
| f(x) |
[1]
Find the zero of f'(x): f'(x) = 0.
Local Maximum, Minimum
Local Maximum, Minimum
[4]
[5]
Find the zero of f''(x): f''(x) = 0.
[6]
[7]
Draw a table like this.
Write the zeros of f'(x) and f''(x), x = 0, 1, 2, in the x row.
Write the zeros of f'(x) and f''(x), x = 0, 1, 2, in the x row.
↓
| x | ... | 0 | ... | 1 | ... | 2 | ... |
|---|---|---|---|---|---|---|---|
| f'(x) | + | 0 | - | - | - | 0 | + |
| f''(x) | - | - | - | 0 | + | + | + |
| f(x) |
↓
| x | ... | 0 | ... | 1 | ... | 2 | ... |
|---|---|---|---|---|---|---|---|
| f'(x) | + | 0 | - | - | - | 0 | + |
| f''(x) | - | - | - | 0 | + | + | + |
| f(x) | ⤶ | ⤶ | ⤷ | ⤷ |
[8]
f'(x): (+), f''(x): (-)
→ f'(x): 3, 2, 1, ... (= slope of f(x))
→ f(x): ⤶
f'(x): (-), f''(x): (-)
→ f'(x): -1, -2, -3, ...
→ f(x): ⤶
f'(x): (-), f''(x): (+)
→ f'(x): -3, -2, -1, ...
→ f(x): ⤷
f'(x): (+), f''(x): (+)
→ f'(x): 1, 2, 3, ...
→ f(x): ⤷
→ f'(x): 3, 2, 1, ... (= slope of f(x))
→ f(x): ⤶
f'(x): (-), f''(x): (-)
→ f'(x): -1, -2, -3, ...
→ f(x): ⤶
f'(x): (-), f''(x): (+)
→ f'(x): -3, -2, -1, ...
→ f(x): ⤷
f'(x): (+), f''(x): (+)
→ f'(x): 1, 2, 3, ...
→ f(x): ⤷
↓
f(x) = x3 - 3x2 + 6
f(0) = 03 - 3⋅02 + 6
= 6
f(1) = 13 - 3⋅12 + 6
= 1 - 3⋅1 + 6
= 7 - 3
= 4
f(2) = 23 - 3⋅22 + 6
= 8 - 3⋅4 + 6
= 14 - 12
= 2
1. Local maximum: (0, 6)
Local minimum: (2, 2)
2. Inflection point: (1, 4) - [10]
f(0) = 03 - 3⋅02 + 6
= 6
f(1) = 13 - 3⋅12 + 6
= 1 - 3⋅1 + 6
= 7 - 3
= 4
f(2) = 23 - 3⋅22 + 6
= 8 - 3⋅4 + 6
= 14 - 12
= 2
| x | ... | 0 | ... | 1 | ... | 2 | ... |
|---|---|---|---|---|---|---|---|
| f'(x) | + | 0 | - | - | - | 0 | + |
| f''(x) | - | - | - | 0 | + | + | + |
| f(x) | ⤶ | 6 | ⤶ | 4 | ⤷ | 2 | ⤷ |
1. Local maximum: (0, 6)
Local minimum: (2, 2)
2. Inflection point: (1, 4) - [10]
[9]
Draw y = f(x) by using the table.
[10]
x = 1
f''(x): - → 0 → +
→ Inflection point: (1, 4)
f''(x): - → 0 → +
→ Inflection point: (1, 4)
Close