Using the First Derivative Test to Analyze Local Minimums in Function f(x)

First Derivative Test: f(x) has a local Minimum at x=c when f'(x)

To determine if a function f(x) has a local minimum at a specific point c, we can use the First Derivative Test

To determine if a function f(x) has a local minimum at a specific point c, we can use the First Derivative Test. The First Derivative Test helps us analyze the behavior of the function by examining the sign of the derivative in the interval around c.

The First Derivative Test states:
1. If f'(x) > 0 for all x in an interval (a, c), and f'(x) < 0 for all x in an interval (c, b), then f(x) has a local minimum at x = c. 2. If f'(x) < 0 for all x in an interval (a, c), and f'(x) > 0 for all x in an interval (c, b), then f(x) has a local maximum at x = c.

Let’s break down each condition of the First Derivative Test for finding a local minimum:

Condition 1: f'(x) > 0 for all x in (a, c), and f'(x) < 0 for all x in (c, b). When f'(x) > 0 for x in (a, c), it means that the function is increasing in that interval, indicating a positive slope. This tells us that the function is moving upward as x approaches c.

When f'(x) < 0 for x in (c, b), it means that the function is decreasing in that interval, indicating a negative slope. This tells us that the function is moving downward as x moves away from c. Based on this behavior, we can infer that the function f(x) has a local minimum at x = c. This means that the point (c, f(c)) is the lowest point on the graph in the vicinity of c. It's important to note that the First Derivative Test is applicable when the function is differentiable on the interval (a, b), and c is a critical point of the function. To evaluate this test, you need to compute the derivative of f(x), f'(x), find the critical points by solving f'(x) = 0 or f'(x) undefined, and then analyze the signs of f'(x) on the intervals (a, c) and (c, b). By applying the First Derivative Test, you can determine if a function has a local minimum at a specific point c.

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