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) = x² - 10 on [-2, 3]

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First, understand that absolute extrema refer to the highest and lowest values a function can take on a given interval. We need to find these values for the function ƒ(x) = x² - 10 on the interval [-2, 3].

To find the absolute extrema, we need to evaluate the function at critical points and endpoints. Start by finding the derivative of ƒ(x). The derivative, ƒ'(x), is obtained by differentiating ƒ(x) = x² - 10, which gives ƒ'(x) = 2x.

Set the derivative equal to zero to find critical points: 2x = 0. Solving this equation gives x = 0. This is a critical point within the interval [-2, 3].

Evaluate the function ƒ(x) at the critical point and at the endpoints of the interval. Calculate ƒ(-2), ƒ(0), and ƒ(3). These values will help determine the absolute maximum and minimum.

Compare the values obtained from ƒ(-2), ƒ(0), and ƒ(3). The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval [-2, 3].

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주요 개념

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Absolute Extrema

Absolute extrema refer to the highest and lowest values of a function over a specified interval. To find these values, one must evaluate the function at critical points, where the derivative is zero or undefined, as well as at the endpoints of the interval. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.

Critical Points

Critical points are values of the independent variable where the derivative of the function is either zero or does not exist. These points are essential in determining the behavior of the function, as they can indicate potential locations for local maxima or minima. In the context of finding absolute extrema, critical points must be evaluated alongside the endpoints of the interval.

Closed Interval

A closed interval, denoted as [a, b], includes all numbers between a and b, including the endpoints a and b themselves. In calculus, analyzing functions over closed intervals is crucial for finding absolute extrema, as it ensures that both the endpoints and any critical points within the interval are considered. This guarantees a comprehensive evaluation of the function's behavior across the entire interval.

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) = x² - 10 on [-2, 3]