Problem 32 - Calculus of a Single Variable

Question

Finding Extrema on a Closed Interval In Exercises \(17-36\) , find the absolute extrema of the function on the closed interval. $$ h(x)=[[2-x]],[-2,2] $$

Step-by-Step Solution

Answer

The function \(h(x)=2-x\) has its absolute maximum of 4 at \(x=-2\) and absolute minimum of 0 at \(x=2\) on the interval [-2,2].

Step 1: Identify the Function and Interval

The given function is \(h(x) = 2-x\) and the closed interval is \([-2,2]\). The goal is to find the absolute maximum and minimum of \(h(x)\) over this interval.

Step 2: Calculate the Derivative

The derivative of \(h(x)\) is used to find any possible local extrema within the interval. The derivative of \(h(x)=2-x\) is \(h'(x)=-1\). However, the derivative is constant, which means there are no local extrema within the interval as the function neither increases or decreases at any point within the interval.

Step 3: Evaluate the Endpoints

As there are no local extrema within the interval based on step 2, we need to check the endpoints of the interval for possible absolute extrema. The two points to check are \(x=-2\) and \(x=2\). The function value at these two points are \(h(-2)= 2-(-2)=4\) and \(h(2)=2-2=0\).

Step 4: Identify the Extrema

With the function evaluations: \(h(-2)=4\) and \(h(2)=0\), we can now identify the absolute extrema. The maximum function value is 4, which occurs at \(x=-2\), and the minimum function value is 0, which occurs at \(x=2\). Thus, the function \(h(x)\) has absolute maximum at \(x=-2\) and absolute minimum at \(x=2\) within the interval [-2, 2].

Key Concepts

Calculus

Calculus is an essential branch of mathematics focused on the study of change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. One of the primary tools of calculus is the derivative, which gives us a way to determine the rate of change of a function at any given point. Moreover, calculus allows us to find extrema, which are the maximum and minimum values a function may take. In applied contexts, these extrema are often crucial as they could represent the most efficient outcomes, the peak stress in a material, or the minimum cost in an optimization problem. Through techniques like closed interval analysis, calculus gives us a systematic approach for locating these important values.

Absolute Extrema

The concept of absolute extrema refers to the highest or lowest points over an entire domain of a function. In practical terms, the absolute maximum is the highest y-value a function attains, and the absolute minimum is the lowest y-value. Unlike local extrema, which are only the highest or lowest points within a small neighborhood of a function, absolute extrema are concerned with the entire interval or domain. Finding these values is fundamental in various fields, including physics, engineering, and economics, as they can represent points of greatest efficiency, output, or profit.

Derivative

The derivative is a powerful concept in calculus, often denoted as \( f'(x) \) for a function \( f(x) \). It represents the slope of the tangent line to the curve defined by the function at a particular point. The derivative measures how a function's output value changes as its input value changes, providing instantaneous rates of change. It enables us to identify where a function is increasing or decreasing, and consequently, where it might have local extrema, which are the peaks and valleys within a specific range. In the process of finding absolute extrema, the derivative can reveal critical points where these local extrema might occur.

Closed Interval Analysis

Closed interval analysis is a technique used in calculus to determine the absolute extrema of a function on a specified closed interval, denoted as \[a, b\]. This interval includes both its endpoints, meaning the function is defined and continuous at these points. The method involves several steps, which usually include: finding the derivative of the function to locate critical points within the interval, evaluating the function at these critical points, and then evaluating the function at the endpoints of the interval. After comparing these values, the highest response is the absolute maximum, and the lowest response is the absolute minimum for the interval. Closed interval analysis is particularly useful because many real-life scenarios have natural constraints that create bounded intervals in which we are interested in finding optima.