Relative Min ( 1st derivative test)
f'(x)=0sign chart for f'(x){1st Derivative test}changes sign negative to positive = MIN
The relative minimum point of a function can be found using the first derivative test. The first derivative test is a method of determining if a critical point (where the derivative is zero or undefined) on a function is a relative maximum, relative minimum, or neither.
To find a relative minimum using the first derivative test, follow these steps:
1. Take the first derivative of the function.
2. Find the critical points by setting the first derivative equal to zero and solving for x.
3. Create a sign chart for the first derivative, which shows whether the function is increasing or decreasing on each interval between critical points.
4. Determine if the function is decreasing before the critical point and increasing afterward. If so, the critical point is a relative minimum.
For example, let’s find the relative minimum of the function f(x) = x^3 – 3x^2 + 2x + 5.
1. Take the first derivative: f'(x) = 3x^2 – 6x + 2.
2. Find the critical points by setting f'(x) = 0: 3x^2 – 6x + 2 = 0. Solving for x, we get x = (3 ± sqrt(3))/3.
3. Create a sign chart for f'(x):
Interval | f'(x)
—————-
(-infinity, (3-sqrt(3))/3) | –
((3-sqrt(3))/3, (3+sqrt(3))/3) | +
((3+sqrt(3))/3, infinity) | –
4. Since f'(x) is negative before (3-sqrt(3))/3 and positive afterward, this critical point is a relative minimum.
Therefore, the relative minimum of f(x) is at x = (3-sqrt(3))/3, and the minimum point is ( (3-sqrt(3))/3, f( (3-sqrt(3))/3 ) ).
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