Problem 31 - Calculus for Biology and Medicine

Question

31\. Graph $$ f(x)=(1-|x|)^{2}, \quad-1 \leq x \leq 2 $$ and determine all local and global extrema on \([-1,2]\).

Step-by-Step Solution

Answer

Local maxima at \(x=0\) and \(x=2\) (value 1), local minima at \(x=-1\) and \(x=1\) (value 0). Global maximum: 1, global minimum: 0.

Step 1: Understand the Function and Domain

The function given is \( f(x) = (1 - |x|)^2 \) and the domain is \([-1, 2]\). The function involves an absolute value, which means we should consider different expressions for \( x \) greater than, less than, and equal to zero.

Step 2: Define the Piecewise Function

For \(-1 \leq x < 0\), \(|x| = -x\), so the function is \( f(x) = (1 - (-x))^2 = (1 + x)^2 \). For \(0 \leq x \leq 2\), \(|x| = x\), so the function simplifies to \( f(x) = (1 - x)^2 \).

Step 3: Plot the Graph

For \(-1 \leq x < 0\), sketch \( f(x) = (1 + x)^2 \), which is a parabola opening upwards with vertex at \((-1, 0)\). For \(0 \leq x \leq 2\), sketch \( f(x) = (1 - x)^2 \), another upward parabola with vertex at \((1, 0)\). The graph is continuous and smooth at \(x=0\) because the function values and their first derivatives match.

Step 4: Calculate the Derivatives

Calculate derivative for \(-1 \leq x < 0\), \( f'(x) = 2(1+x) \). For \(0 \leq x \leq 2\), \( f'(x) = 2(x-1) \). These will help identify critical points where \( f'(x) = 0 \).

Step 5: Find Critical Points

Setting \( f'(x) = 0 \) for both expressions gives us the points \(x = -1\) for \((-1 \leq x < 0)\) and \(x = 1\) for \((0 \leq x \leq 2)\). No critical points in the region \(-1 < x < 0\) or \(0 < x < 2\).

Step 6: Evaluate the Function at Critical and Boundary Points

Evaluate \( f(x) \) at \(x = -1\), \(x = 0\), \(x = 1\), and \(x = 2\). We have \( f(-1) = 0 \), \( f(0) = 1\), \( f(1) = 0 \), and \( f(2) = 1\).

Step 7: Identify Local and Global Extrema

From evaluations, \( f(x) \) has a local maximum at \(x=0\) and \(x=2\) with value 1, and a local minimum at \(x=-1\) and \(x=1\) with value 0. Thus, the global maximum on \([-1, 2]\) is 1 and the global minimum is 0.

Key Concepts

Graphing

Graphing functions allows us to visually analyze the behavior and characteristics of the function over a given interval. In this example, we have the function \( f(x) = (1 - |x|)^2 \) with a domain of \([-1, 2]\). To effectively graph this, one must first understand the changes in behavior at key points that affect the shape of the graph.

A significant part of graphing involves identifying critical points where the function changes direction or has a local maximum or minimum. Calculating derivatives helps locate these points. In our exercise, derivatives were used to determine points where \( f'(x) = 0 \), indicating potential extrema or direction changes in the graph.

Lastly, ensuring continuity at transition points, such as \( x = 0 \) where \( |x| \) transitions from \(-x\) to \(x\), is vital. The function must be joined smoothly to avoid gaps or jumps in the graph, indicating areas where the function shifts forms. This analysis ensures the graph is complete and accurate.

Piecewise Functions

Piecewise functions are created by combining different function rules over specific intervals. They are key when dealing with absolute values or conditions over distinct sub-domains.

The function \( f(x) = (1 - |x|)^2 \) is inherently piecewise because the absolute value causes it to split over different expressions based on the input \( x \).

In this case, the function behaves differently in two ranges. For \(-1 \leq x < 0\), the function transforms to \( f(x) = (1 + x)^2 \), indicating a parabola opening upwards with specific behavior on this interval. For \(0 \leq x \leq 2\), the function becomes \( f(x) = (1 - x)^2 \), another upward parabola. Each subfunction caters to changes in \(|x|\), providing a complete description for \(f(x)\) over its domain.

Understanding these subinterval rules is crucial for interpreting the graph and analyzing the behavior of the overall function.

Local and Global Extrema

In calculus, local and global extrema represent the points where a function obtains its maximum or minimum values. Local extrema are specific to small neighborhoods, while global extrema concern the entire interval.

For our task, identifying local extrema involved evaluating the function at critical points found by setting the derivative, \( f'(x) \), equal to zero. For \(-1 \leq x < 0\), this occurs at \(x = -1\), and for \(0 \leq x \leq 2\), at \(x = 1\). These give local minimum points with \( f(-1) = 0 \) and \( f(1) = 0 \).

The function's endpoints, \( x = 0 \) and \( x = 2 \), are also significant as potential global extrema, both reaching a maximum value of \(1\). Hence, with calculations and evaluations, these points end up defining both the local and global behavior of the function, guiding where it peaks or hits low in the interval \([-1, 2]\).

Recognizing these points helps in graph analysis and understanding function behaviour comprehensively.