Identifying Extrema

In Exercises 41–52:

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

f(t) = t³ − 3t², −∞ < t ≤ 3

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First, find the derivative of the function f(t) = t³ − 3t². The derivative, f'(t), will help us identify critical points where local extrema might occur.

Calculate f'(t) by differentiating f(t) = t³ − 3t². Using the power rule, the derivative is f'(t) = 3t² - 6t.

Set the derivative f'(t) equal to zero to find the critical points: 3t² - 6t = 0. Factor the equation to solve for t.

Factor the equation: 3t(t - 2) = 0. This gives the critical points t = 0 and t = 2.

Evaluate the function f(t) at the critical points and at the endpoint t = 3 to determine the local extrema. Compare the values of f(t) at t = 0, t = 2, and t = 3 to identify the local maximum and minimum values within the domain −∞ < t ≤ 3.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema. For the function f(t) = t³ − 3t², finding the derivative and setting it to zero helps identify critical points, which are essential for determining where local maxima or minima might occur.

First Derivative Test

The First Derivative Test is used to determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, we can infer the behavior of the function. This test is crucial for identifying the nature of extrema in the function f(t) = t³ − 3t².

Endpoints in a Domain

When identifying extrema in a given domain, it's important to evaluate the function at the endpoints of the domain as well. Since the domain for f(t) = t³ − 3t² is −∞ < t ≤ 3, we must consider the behavior of the function as t approaches 3, as this endpoint could potentially be a location for an extreme value.

Identifying ExtremaIn Exercises 41–52:a. Identify the function’s local extreme values in the given domain, and say where they occur.f(t) = t³ − 3t², −∞ < t ≤ 3