Determining Local Maxima and Minima | First and Second Derivative Test Explained

Looking at a graph of f(x)… how do you know where F(x) has a local max or min?

To determine where a function f(x) has a local maximum or minimum, you need to examine the first and second derivatives of the function and identify critical points

To determine where a function f(x) has a local maximum or minimum, you need to examine the first and second derivatives of the function and identify critical points.

1. First derivative test:
First, find the derivative of f(x), denoted as f'(x), and locate the critical points by setting f'(x) = 0 or when f'(x) is undefined. These critical points correspond to potential local maxima or minima.

– If f'(x) changes sign from positive to negative at a critical point, it indicates a local maximum at that point.
– If f'(x) changes sign from negative to positive at a critical point, it indicates a local minimum at that point.

2. Second derivative test:
To confirm whether the critical points are local maxima or minima, you can use the second derivative test. Take the second derivative, f”(x), and evaluate it at each critical point.

– If f”(x) > 0 at a critical point, it confirms that the point is a local minimum.
– If f”(x) < 0 at a critical point, it confirms that the point is a local maximum. If the second derivative is zero or if the test is inconclusive (neither positive nor negative), further analysis is needed. Also, keep in mind: - Endpoints of the interval of the graph being analyzed may also act as local maxima or minima. - If the first and/or second derivative is undefined at a point, it's worth investigating further as it may be a point of interest. By utilizing these tests, you can determine where a function f(x) has local maxima or minima based on the behavior of its derivative(s).

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