Problem 64 - University Calculus: Early Transcendentals

Question

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=e^{x}-e^{-x}$$

Step-by-Step Solution

Answer

The function has no extreme values over its domain.

Step 1: 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 \( (-\infty, \infty) \).

Step 2: 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, \( y' = \frac{d}{dx}(e^x) - \frac{d}{dx}(e^{-x}) = e^x + e^{-x} \).

Step 3: 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.

Step 4: 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.

Step 5: 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.

Key Concepts

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.

To locate these extrema, it's essential to analyze the function's behavior by finding its critical points and checking the function's endpoints when on a closed interval. However, in this problem, the function does not yield any critical points, indicating that there aren't local extrema. Also, given that it extends to infinity, absolute extrema don't exist in this infinite domain. Thus, no extreme values are present.

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.

Since the derivative \( y' \) is always positive (because \( e^x + e^{-x} > 0 \) for all real \( x \)), it tells us that the function \( y \) is always increasing.

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.

In simpler terms, without critical points, the function cannot have local maximums or minimums at any point within its domain.

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.

In conclusion, since the function is always increasing and has no critical points, it smoothly extends from negative infinity to positive infinity without any local or absolute extrema within its natural domain of all real numbers.