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Magazine Chapter 4: Problem 64
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=e^{x}-e^{-x}$$
Short Answer
Expert verified
The function has no extreme values over its domain.
Step by step solution
01
Identify the Function and its Domain
The given function is \( y = e^x - e^{-x} \). This function is defined for all real numbers \( x \) because both \( e^x \) and \( e^{-x} \) are defined for all real \( x \). Therefore, the natural domain is \( (-\infty, \infty) \).
02
Find the First Derivative
To find critical points where extreme values might occur, we first need the first derivative of the function, \( y = e^x - e^{-x} \). Calculating, \( y' = \frac{d}{dx}(e^x) - \frac{d}{dx}(e^{-x}) = e^x + e^{-x} \).
03
Set the First Derivative to Zero
To find critical points, solve \( y' = 0 \): \[ e^x + e^{-x} = 0 \].\However, the terms \( e^x \) and \( e^{-x} \) are always positive for real \( x \), and hence their sum cannot be zero. There are no real solutions, so there are no critical points.
04
Analyze the First Derivative
Since \( e^x + e^{-x} > 0 \) for all \( x \), the first derivative \( y' \) is always positive. This means \( y = e^x - e^{-x} \) is a strictly increasing function on its entire domain.
05
Determine Extreme Values
Since \( y' > 0 \) for all \( x \), the function has no local minima or maxima. As \( x \to -\infty, e^x \to 0 \) and \( e^{-x} \to \infty \), so \( y \to -\infty \). As \( x \to \infty, e^x \to \infty \) and \( e^{-x} \to 0 \), so \( y \to \infty \). Thus, there are no absolute extrema within the domain.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Extreme Values
Extreme values in calculus refer to the highest and lowest values that a function can take on a given interval. They can be categorized into two types:
- Absolute extrema: These are the ultimate highest or lowest values a function can attain over its entire domain or on a closed interval.
- Local extrema: These occur at certain points within a domain where the function reaches a peak or valley, higher or lower than all nearby points.
Derivative
The derivative of a function helps us understand how the function's output changes as its input changes. Essentially, it provides the slope of the tangent line to the function at any given point.
- In our exercise, we took the derivative of the function \( y = e^x - e^{-x} \).
- The derivative \( y' \) is found to be \( e^x + e^{-x} \).
- This derivative gives us insights into the function's rate of change and its slope at each point.
Critical Points
Critical points are where a function's derivative is zero or undefined. These points are potential candidates for identifying local extrema.
- To find potential critical points, solve \( y' = 0 \).
- Here, \( e^x + e^{-x} = 0 \), is not possible because both \( e^x \) and \( e^{-x} \) are always positive.
- As there are no solutions to this equation, the function has no critical points.
Function Analysis
Analyzing a function involves understanding its behavior over its domain. This includes determining where the function increases, decreases, and where it might reach its extreme values.
- For \( y = e^x - e^{-x} \), we know from the derivative \( y' = e^x + e^{-x} \) that \( y \) is always increasing.
- There are no turning points or critical points to consider.
- The behavior of the function as \( x \) goes to either positive or negative infinity affects its extreme values.
