Question

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=4 \sqrt{x}-2 x+1, \quad[0,6] $$

Short answer

The absolute extrema of the function \( f(x)=4\sqrt{x}-2x+1 \) on the interval [0,6] can be determined graphically by graphing the function and identifying the highest and lowest points within the interval. The y-values at these points are the absolute maximum and minimum of the function, respectively.

Step-by-step solution

Understand the Function

The function given is \( f(x)=4\sqrt{x}-2x+1 \). It's vital to understand this function well because its nature will determine what the graph looks like and thus, where the extrema will be located.

Graph the Function

Using a graphing utility, plot the function \( f(x)=4\sqrt{x}-2x+1 \) on the interval [0,6]. Pay close attention to the shape of the graph, as well as where it reaches its highest and lowest points within this interval.

Identify the Extrema

Upon observing the graph, identify the highest and lowest points within the interval [0,6]. These points represent the absolute maximum and minimum of the function, respectively. The x-values at which these points occur are the x-values that yield the absolute extrema when plugged into the function \( f(x)=4\sqrt{x}-2x+1 \).

Calculate the Extrema

Substitute the x-values identified in the previous step into the function to find the y-values, i.e., the function values at these points. This will give you the absolute maximum and minimum values of the function on the interval [0,6].

Key concepts

Graphing Utility

The use of a graphing utility is a significant aid in visualizing the behavior of functions, which is especially useful when seeking to identify absolute extrema. These digital tools allow students to accurately render complex functions like \( f(x)=4\sqrt{x}-2x+1 \) without the need for manual calculations and plotting.

For the exercise in question, employing a graphing utility simplifies the process. It enables students to see the curve between the given interval \([0,6]\), highlighting crucial points such as peaks and troughs representing the highest and lowest function values. By inputting the function into the utility, one can quickly generate a graph that makes evident the locations of potential extrema. This immediate visual representation removes ambiguity and allows for more confident identification of important values within the function's range.

Closed Interval Optimization

Closed interval optimization is a procedure used to find the absolute maximum and minimum values of a continuous function over a specific range. This method is vital because it confines the search for extrema to a manageable set of values, which is the closed interval \([a, b]\).

In the context of the given function \( f(x)=4\sqrt{x}-2x+1 \), the closed interval is \([0,6]\). To perform closed interval optimization, one must evaluate the function at critical points within the interval, which includes the endpoints and any points where the derivative equals zero or is undefined. Mathematically, it boils down to comparing function values at these critical points to deduce the absolute minimum and maximum. It's a systematic approach that ensures no possible extrema are overlooked in the interval, provided the function is continuous.

Extrema of a Function

Extrema of a function refer to the points at which a function reaches its minimum or maximum values. There are two types, absolute (or global) extrema and relative (or local) extrema. An absolute maximum or minimum is the highest or lowest value of the function on its entire domain or on a specified interval, while a relative extremum is the highest or lowest point in a neighborhood around a given point in the domain.

For the given exercise, determining the absolute extrema within the interval \([0,6]\) involves pinpointing the x-values that produce the highest (maximum) and lowest (minimum) y-values when plugged into \( f(x)=4\sqrt{x}-2x+1 \). These extremal values are critical in many areas of mathematics, including optimization problems and calculus, as they can tell us a great deal about the function’s behavior and are often used to solve practical problems in engineering, physics, economics, and more.