Problem 7 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ , each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extreme- value theorem. With the help of a graphing calculator or spreadsheet, graph each function and locate its global extrema. (Note that a function may have more than one global minimum or maximum point.) $$ f(x)=e^{-|x|},-1 \leq x \leq 1 $$

Step-by-Step Solution

Answer

Global maximum at \((0, 1)\), global minima at \((-1, 0.3679)\) and \((1, 0.3679)\).

Step 1: Understand the Function

The given function is \( f(x) = e^{-|x|} \) defined on the closed interval \([-1, 1]\). The function involves both the exponential function and the absolute value of \(x\). \( e^{-|x|} \) is a continuous function across its domain, which is suitable for applying the extreme-value theorem.

Step 2: Analyze the Function Behavior

The absolute value in the exponent, \( -|x| \), means that for negative and positive values of \(x\), the calculation is the same magnitude but opposite signs. The expression \( e^{-|x|} \) decreases from 1 at \(x=0\) to approximately \(0.3679\) at \(x=\pm 1\). Thus, the function is symmetric about the \(y\)-axis.

Step 3: Graph the Function

Using a graphing calculator or spreadsheet, plot \( f(x) = e^{-|x|} \) over the interval \([-1, 1]\). The graph shows a peak at \(x = 0\) because the exponential function yields a maximum value when the exponent is 0, and steadily decreases towards both \(-1\) and \(1\).

Step 4: Locate the Global Extrema

From the graph, observe that the highest point (global maximum) is at \(x = 0\) where \(f(0) = e^{0} = 1\). The lower bound values \(x = -1\) and \(x = 1\) yield \(f(-1) = f(1) = e^{-1} \approx 0.3679\), both being equal and considered global minima, as the function takes its smallest value there over the interval.

Key Concepts

Understanding Continuous Functions

When we talk about a continuous function, we refer to a function where there are no breaks, jumps, or holes in its graph on a given interval. This means you could draw the function on a graph without lifting your pencil from the paper.
Continuous functions are crucial because the extreme-value theorem relies on them. This theorem states that if a function is continuous over a closed interval \([a, b]\), then it must attain both a maximum and a minimum value.
For example, in our exercise, the function \(f(x) = e^{-|x|}\) is continuous over the closed interval \([-1, 1]\). This allows us to use the extreme-value theorem confidently, knowing the function will hit both a highest and lowest point in this interval.

Exploring Global Extrema

Global extrema are the overall highest and lowest points on a function's graph over its defined interval. These points correspond to the global maximum and minimum values.
For the function \(f(x) = e^{-|x|}\), the global maximum is at \(x = 0\) since \(f(0) = e^{0} = 1\) is the highest value on the interval \([-1, 1]\).
On the other hand, the global minimum can exist in multiple places if they yield the same value. In this case, the boundaries \(x = -1\) and \(x = 1\) both give \(f(-1) = f(1) = e^{-1} \approx 0.3679\). Hence, these points are the locations of the global minima as they represent the lowest values the function reaches on the specified interval.

Delving into Absolute Value

The absolute value concept measures how far a number is from zero, regardless of its direction on the number line. It's always non-negative.
For instance, in the function \(f(x) = e^{-|x|}\), the expression \(-|x|\) affects the behavior of the function by symmetrizing it about the \y\-axis.
Due to the absolute value, \(f(x)\) will take the same value at both \(x\) and \(-x\), leading to a graph that's mirrored on the left and right of the \y\-axis. Understanding absolute value is vital because it often simplifies the behavior analysis of symmetric functions like this one.

Using a Graphing Calculator

A graphing calculator is a valuable tool for visualizing functions and identifying extrema. It allows you to plot the function easily and see its behavior over a given interval.
For example, using a graphing calculator with the function \(f(x) = e^{-|x|}\), you can accurately plot the curve in the domain \([-1, 1]\).
This tool will help you spot the peak at \(x = 0\), confirming it's the global maximum, and also demonstrate the function's decrease to the same values at both endpoints of the interval. A graphing calculator helps bridge the gap between theoretical understanding and visual intuition.