Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x³ - 3x² on [-1, 3]

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First, understand that absolute extrema refer to the highest and lowest values of a function on a given interval. To find these, we need to evaluate the function at critical points and endpoints of the interval.

Find the derivative of the function ƒ(x) = x³ - 3x². The derivative, ƒ'(x), will help us identify critical points where the slope is zero or undefined. Calculate ƒ'(x) = 3x² - 6x.

Set the derivative ƒ'(x) = 0 to find critical points. Solve the equation 3x² - 6x = 0. Factor the equation to get 3x(x - 2) = 0, which gives the critical points x = 0 and x = 2.

Evaluate the function ƒ(x) at the critical points and at the endpoints of the interval [-1, 3]. Calculate ƒ(-1), ƒ(0), ƒ(2), and ƒ(3) to find the values of the function at these points.

Compare the values obtained from evaluating the function at the critical points and endpoints. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval [-1, 3].

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주요 개념

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Critical Points

Critical points are values of x in the domain of a function where the derivative is either zero or undefined. These points are essential for finding local maxima and minima, as they indicate where the function's slope changes. To locate absolute extrema, one must first find these critical points within the given interval.

Endpoints of the Interval

When determining absolute extrema on a closed interval, it is crucial to evaluate the function at both the critical points and the endpoints of the interval. The absolute maximum or minimum could occur at these endpoints, so they must be included in the analysis to ensure all potential extreme values are considered.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function around that point. This test helps in identifying the nature of the extrema found in the analysis.

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.ƒ(x) = x³ - 3x² on [-1, 3]