Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = (x + 1)², −∞ < x ≤ 0

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First, understand that local extrema refer to the local minimum or maximum values of a function within a given domain. For the function f(x) = (x + 1)^2, we need to find these values for -∞ < x ≤ 0.

To find the local extrema, we first need to find the derivative of the function f(x). The derivative, f'(x), will help us identify critical points where the slope of the tangent is zero or undefined.

Calculate the derivative of f(x) = (x + 1)^2. Using the power rule, the derivative is f'(x) = 2(x + 1).

Set the derivative equal to zero to find the critical points: 2(x + 1) = 0. Solve for x to find the critical point.

Evaluate the function f(x) at the critical point and at the endpoints of the domain (if applicable) to determine the local extrema. Since the domain is -∞ < x ≤ 0, consider the endpoint x = 0 as well.

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주요 개념

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Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific interval. These are points where the function changes direction, and can be identified by analyzing the derivative of the function. In the context of the given domain, local extrema are found by evaluating the function's behavior at critical points and endpoints.

Derivative and Critical Points

The derivative of a function provides information about its rate of change and helps identify critical points where the derivative is zero or undefined. These points are potential locations for local extrema. For the function f(x) = (x + 1)², the derivative f'(x) = 2(x + 1) can be used to find critical points by setting it equal to zero and solving for x.

Domain Considerations

The domain of a function specifies the set of input values for which the function is defined. In this problem, the domain is −∞ < x ≤ 0, which affects where extrema can occur. It's crucial to consider endpoints and any restrictions in the domain when identifying extrema, as they can influence the function's behavior and the location of extrema.

Identifying ExtremaIn Exercises 41–52:a. Identify the function’s local extreme values in the given domain, and say where they occur.f(x) = (x + 1)², −∞ < x ≤ 0