Mastering The First Derivative Test: Analyzing Critical Points Of Functions

First Derivative Test

For a continuous function f.(1)if f’ changes sign from positive to negative at a critical point c, then f has a local maximum value at c;(2)if f’ changes sign from negative to positive at a critical point c, then f has a local minimum value at c;(3)if f’ does not change sign at a critical point c, then f has no local extreme value at c;(4)if f’ < 0 (f' > 0) for x > a where a is a left endpoint in the domain of f, then f has a local maximum (minimum) value at a;(5)if f’ < 0 (f' > 0) for x < b where b is a right endpoint in the domain of f, then f has a local minimum (maximum) value at b.

The first derivative test is a method used to analyze the behavior of a function at certain critical points by examining the sign of the derivative of the function. By studying the sign of the first derivative, we can determine the nature of the critical point (maximum or minimum) and whether the critical point is a local or a global extremum.

To apply the first derivative test, we need to follow these steps:

1. Find the critical points of the function by finding the values of x where f'(x) = 0 or f'(x) does not exist.
2. Determine the sign of the derivative to the left and right of each critical point. We can use a sign chart or test points to do this.
3. If the sign of the derivative changes from positive to negative, the critical point is a local maximum. If it changes from negative to positive, the critical point is a local minimum.
4. If the sign of the derivative doesn’t change (remains positive or negative) from left to right, the critical point is either a point of inflection, a horizontal tangent, or a global extremum.

It’s important to note that the first derivative test only applies to continuous functions.

Overall, the first derivative test is a useful tool for analyzing the behavior of a function and determining the nature of critical points.

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