Find the relative maxima and relative minima, if any, of each function. f(x)=(x-1)^2 / 3

Name: Problem 74, Chapter 4: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-18T19:39:22-08:00
Duration: 1 min 9 s
Channel: Tyler Moulton
Description: Find the relative maxima and relative minima, if any, of each function. f(x)=(x-1)^2 / 3

step 1 We want to find the relative maxima and minima for f equals x minus 1 to the power of 2 thirds.

Find the relative maxima and relative minima, if any, of...

Key Concepts

Critical Points

Critical points are values in the domain of a function where its derivative is either zero or does not exist. These points are important because they are potential locations for local extrema, such as relative maxima and minima. Identifying critical points is the first step in analyzing the behavior of functions in calculus.

Relative Extrema

Relative extrema refer to the points in the function's domain where the function attains a local maximum or minimum compared to its nearby values. A relative maximum is a point where the function value is higher than those of all other points in its immediate vicinity, while a relative minimum is where it is lower. These concepts are central in optimization and understanding the local behavior of functions.

Differentiability and Domain

Differentiability concerns whether a function has a derivative at a certain point, which often requires that the function is smooth at that point. The domain must be analyzed to understand where the function and its derivative are defined. Points where the derivative does not exist need special attention, as they can still be candidates for relative extrema.

First Derivative Test

The first derivative test is a method used to determine the nature of a critical point by analyzing the change in sign of the derivative around that point. If the derivative changes from negative to positive at a point, it indicates a relative minimum, while a change from positive to negative signifies a relative maximum. This test is a fundamental tool in calculus for classifying the behavior of functions around critical points.

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