Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{5}-20 x+5\) on \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The function has two critical points: \(-\sqrt[4]{4}\) and \(\sqrt[4]{4}\). \(-\sqrt[4]{4}\) is a local maximum and \(\sqrt[4]{4}\) is a local minimum. Because the function is defined on the interval \((-\infty, \infty)\), the maximum and minimum at these points are also the absolute maximum and minimum.

Step 1: Derive Function

First differentiate the function to find its derivative. For the function \(f(x)=x^{5}-20 x+5\), the derivative \(f'(x) = 5x^{4}-20\).

Step 2: Find Critical Points

To find the critical points, set the first derivative equal to zero and solve for x. \(0 = 5x^{4}-20\), which simplifies to \(x^{4} = 4\). This gives four critical points: \(x = -\sqrt[4]{4}, -\sqrt[4]{4}, \sqrt[4]{4}, \sqrt[4]{4}\).

Step 3: Classify Critical Points

For each critical point, examine the sign of the first derivative to the immediate left and right of the point. If the first derivative changes from negative to positive, the function is increasing and the point is a local minimum. If the derivative changes from positive to negative, the function is decreasing and the point is a local maximum. For the critical points here, \(x = -\sqrt[4]{4}\) and \(x = \sqrt[4]{4}\) the first derivative changes sign, indicating these points are local extremums. The other two critical points have the same sign, indicating neither increase nor decrease at these points.

Step 4: Identify Maximum and Minimum Values

Evaluate the function at the critical points \(-\sqrt[4]{4}\) and \(\sqrt[4]{4}\) to find the corresponding y-values. For \(f(-\sqrt[4]{4})\) and \(f(\sqrt[4]{4})\), the outputted y-values are the minimum and maximum values of the function respectively on the interval \((-\infty, \infty)\).

Key Concepts

Local Maximum

A local maximum is a point on a function where the value of the function is higher than at any nearby points. Think of it as the peak of a hill. To identify a local maximum, you need to examine the function's behavior around the critical points (points where the derivative equals zero or is undefined). If the slope of the tangent (represented by the derivative) changes from positive (going up) to negative (going down), then you've found a local maximum. In the exercise example, we find such points by examining the first derivative of the function, which changes signs at the critical points. By checking the sign of the derivative before and after these critical points, we confirm whether they are local maximums. This change in sign represents the hill's peak in the function's graph.

Local Minimum

A local minimum represents a point on a function where the function value is lower than that of surrounding points – akin to the bottom of a valley. To find a local minimum, check the derivative's behavior (the slope) at and around critical points. When the derivative goes from negative (downhill) to positive (uphill), it signals a local minimum. In our example, analyzing the derivative function reveals where the tangent slope shifts from negative to positive. This change points out the valleys of the curve, helping us identify the function’s low points. Such transitions in slope are crucial in determining the local minima and are used alongside local maxima when classifying function points.

Derivative

The derivative of a function tells us how the function's values are changing at any point. It's essentially the slope of the function at any given point. Knowing the derivative of a function enables us to find critical points which give us clues about the function's behavior. To find a derivative, use differentiation rules. For example, in the exercise, the function given is differentiated to form a new function, indicating how steep the slopes are at any point along the graph. This derivative helps in unlocking critical points by setting it to zero and solving. Understanding derivatives is crucial as they guide us in pinpointing where a function's graph levels out (critical points) and if it corresponds to a peak or a valley. Mastery of derivatives opens the door to deeper insights into function behaviors beyond just basic curve appearances.

Function Classification

Function classification involves determining the key characteristics and features of a function, especially focusing on the extremes like maximum and minimum points. Classification starts with finding the derivative and setting it to zero to locate critical points – where functions enjoy unique characteristics like being maxima or minima at those spots. These points are central to understanding how a function behaves. Classifying a function helps us see a bigger picture of how its values change over given intervals. For instance, our exercise entails classifying whether critical points are peaks or valleys, indicating how often a function achieves its highest or lowest points along its domain. Awareness of these classifications helps predict the function's trends, making them a fundamental concept in calculus and real-world applications.