Explanation
Extremum points are special points of a function where the slope 0 is zero. There, the function has either a maximum or a minimum. To find extremum points, you first set the first derivative of the function equal to 0 and then check with the second derivative or the sign change whether it is a maximum or minimum. The fundamental formula for calculating extremum points is:
Formula for calculating extremum points:
with a sign change
Procedure
To calculate extremum points, follow these steps:
1. Take the first derivative of the function.
2. Set the first derivative equal to 0 and solve for x solve.
3. Take the second derivative and substitute the found x-values.
4. If the second derivative is positive, there is a minimum. If it is negative, there is a maximum. Alternatively, you can also examine the sign change of the first derivative.
Examples
Determine the extremum points of the function f(x) = x^2.
1. Derivative: f'(x) = 2x. Set f'(x) = 0 → x = 0.
2. Derivative: f''(x) = 2. Since f''(0) = 2 > 0, lies at x = 0 a minimum. The minimum is (0\,|\,0).
Determine the extremum points of the function f(x) = x^3 - 3x.
1. f'(x) = 3x^2 - 3 = 0 \Rightarrow x_1 = 1,\, x_2 = -1.
2. f''(x) = 6x. f''(1) = 6 > 0 → Minimum at x = 1. f''(-1) = -6 < 0 → Maximum at x = -1. Minimum (1\,|\,-2), Maximum (-1\,|\,2).
Note
- ★Extremum points are points where the slope 0 is.
- ★The second derivative or the sign change of the first derivative shows whether there is a maximum or minimum.
Exercises
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