Problem 5 - Calculus for Biology and Medicine

Question

Each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extremevalue theorem. With the help of a graphing calculator, graph each function and locate its global extrema. (Note that a function may assume a global extremum at more than one point.) \(f(x)=|x|,-1 \leq x \leq 1\)

Step-by-Step Solution

Answer

Global minimum: 0 at \(x=0\). Global maximum: 1 at \(x=-1\) and \(x=1\).

Step 1: Understand the Function

The function given is \( f(x) = |x| \). This is an absolute value function and is defined for the interval \(-1 \leq x \leq 1\). The function is continuous over this interval and satisfies the conditions of the extreme value theorem, which implies it will have both a global maximum and a global minimum.

Step 2: Graph the Function

Use a graphing calculator to graph \( f(x) = |x| \). Observe that the graph is a V-shaped curve with the vertex at the origin \((0,0)\). The left arm descends from \((-1,1)\) to \((0,0)\) and the right arm ascends from \((0,0)\) to \((1,1)\).

Step 3: Identify Critical Points

Identify critical points within the domain \([-1,1]\). The critical point in the absolute value function \( f(x) = |x| \) occurs at \( x = 0 \) (the vertex of the V-shape) because this is where the function changes direction. The endpoints \( x = -1 \) and \( x = 1 \) are also important to consider.

Step 4: Determine Function Values at Critical Points and Endpoints

Evaluate the function at the critical point and endpoints. - At \( x = -1 \), \( f(-1) = |-1| = 1 \).- At \( x = 0 \), \( f(0) = |0| = 0 \).- At \( x = 1 \), \( f(1) = |1| = 1 \).

Step 5: Identify Global Extrema

Based on the function values:- The global minimum is \(0\) at \( x = 0 \).- The global maximum is \(1\), which occurs at both \( x = -1 \) and \( x = 1 \). Therefore, the function has a global minimum at the origin and a global maximum at the endpoints of the interval.

Key Concepts

Absolute Value Function

The absolute value function, represented as \(f(x) = |x|\), is one of the most fundamental functions in mathematics. It's defined as the distance of a number from zero on the number line, regardless of direction. Therefore, its value is always non-negative. The absolute value of a number \(x\) is described as:

• If \(x > 0\), then \(|x| = x\).
• If \(x = 0\), then \(|x| = 0\).
• If \(x < 0\), then \(|x| = -x\).

This function is characterized by its V-shaped graph, which has a vertex at the origin (0, 0) when plotted. The two arms of the "V" extend symmetrically in both positive and negative x-directions. The domain of \(f(x) = |x|\) is all real numbers, and its range is all non-negative real numbers (\(y \geq 0\)).
Understanding the absolute value function is crucial because it appears often in solving real-world problems such as calculating distances and inequalities.

Global Extrema

In the context of functions, global extrema (extremes) refer to the highest or lowest values that a function attains over its entire domain. The global maximum is the highest overall point on a graph, whereas the global minimum is the lowest.
When applying the Extreme Value Theorem, if a function is continuous on a closed interval \( [a, b] \), it guarantees the existence of a global maximum and a global minimum within that interval. For the absolute value function \(f(x) = |x|\) over the interval \([-1, 1]\):

• At the endpoint \(x = -1\), the function value is \(f(-1) = 1\).
• The function reaches its global minimum value of \(0\) at the vertex \(x = 0\).
• At the endpoint \(x = 1\), the function value is \(f(1) = 1\).

Both \(x = -1\) and \(x = 1\) are locations for the global maximum value of \(1\), indicating the function tops its peak at more than one point.
Understanding global extrema helps in analyzing the behavior of a function over an interval.

Graphing Calculator

Graphing calculators are powerful tools in mathematics education that help visualize and understand functions like the absolute value function \(f(x) = |x|\). By inputting the function into a graphing calculator and plotting the graph, students can see the V-shaped curve that represents the function visually. This helps in learning how different equations change the shape and position of graphs.
A graphing calculator allows users to:

• Plot graphs over specific intervals.
• Identify the nature of key points, such as minima and maxima.
• Explore symmetry, behavior, and transformations of graphs.

For the absolute value function in the interval \([-1, 1]\), students can see that while the graph is smooth and continuous, it forms a V-shape with peaks (global maxima) at the endpoints and a dip (global minimum) at the vertex. Using a graphing calculator not only makes solving mathematical problems more interactive but also deepens the understanding of concepts like extrema and continuity in functions.