Explanation
An inflection point is a point on the graph of a function where the curvature changes. At this point, the curvature is zero and the second derivative of the function changes its sign. To determine inflection points, set the second derivative of the function equal to zero and then check the sign change or use the third derivative.
The general formula for determining inflection points is:
Procedure
To determine inflection points, follow these steps:
1. Calculate the first derivative f'(x).
2. Calculate the second derivative f''(x).
3. Set the second derivative equal to zero and solve for x at.
4. Check with the third derivative f'''(x), if it is nonzero at the found points. If the third derivative is zero, there is no inflection point.
Alternatively, you can also check the sign change of the second derivative.
Examples
Determine the inflection point of the function f(x) = x^3.
1. Calculate derivatives: f'(x) = 3x^2,\, f''(x) = 6x,\, f'''(x) = 6.
2. Set the second derivative equal to zero: 6x = 0 \Rightarrow x = 0.
3. Check the third derivative: f'''(0) = 6 \neq 0 ⇒ Inflection point at x = 0.
4. Calculate y-value: f(0) = 0 ⇒ Inflection point (0\,|\,0).
Determine the inflection points of the function f(x) = x^4 - 4x^3.
1. Calculate derivatives: f''(x) = 12x^2 - 24x,\, f'''(x) = 24x - 24.
2. Set the second derivative equal to zero: 12x(x-2) = 0 \Rightarrow x_1 = 0,\, x_2 = 2.
3. Check the third derivative: Inflection points at x = 0 and x = 2.
4. Calculate y-values: Inflection points (0\,|\,0) and (2\,|\,-16).
Note Box
- ★Inflection points are located where the curvature is zero and the second derivative changes its sign.
- ★The third derivative must not be zero at an inflection point.
Exercises
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