Problem 27 - Thomas Calculus in SI Units

Question

Suppose that \(f(-1)=3\) and that \(f^{\prime}(x)=0\) for all \(x .\) Must \(f(x)=3\) for all \(x ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, \(f(x) = 3\) for all \(x\) because \(f'(x) = 0\) means \(f(x)\) is constant, and \(f(-1) = 3\) sets that constant.

Step 1: Understand the given information

We are given that the function value at a specific point is \(f(-1) = 3\), and that the derivative of the function \(f'(x) = 0\) for all values of \(x\). We need to determine what these two pieces of information imply about the function \(f(x)\).

Step 2: Analyze the derivative condition

Since \(f'(x) = 0\) for all \(x\), this means that the rate of change or the slope of the function \(f(x)\) is zero everywhere. This suggests that \(f(x)\) is a constant function. Recall that a constant function has the same value for every \(x\).

Step 3: Use initial condition to find the constant value

Given \(f(-1) = 3\), and knowing that \(f(x)\) is constant (from Step 2), we can conclude that this constant function has the value 3 everywhere. Thus, \(f(x) = 3\) for all \(x\).

Step 4: Conclusion

Since both the derivative and the initial point imply that the function does not change its value anywhere, indeed \(f(x) = 3\) for all \(x\). Therefore, the condition that \(f'(x) = 0\) necessitates \(f(x)\) being a constant, and the initial condition provides the constant value.

Key Concepts

Derivative and Its Meaning

A derivative, represented by \( f'(x) \), is a key concept in calculus that measures how a function changes as its input changes. Specifically, it describes the rate of change or the slope of the function at any given point.

• If the derivative is positive, the function is increasing; if negative, the function is decreasing.
• If the derivative is zero, the function is either flat at that point or across intervals.

In our original exercise, we are given that \( f'(x) = 0 \) for all \( x \). This implies that the slope of the function is zero everywhere, meaning that the function does not rise or fall as \( x \) changes. As a result, \( f(x) \) must be constant, since any change in \( x \) does not result in a change in \( f(x) \). Such behavior characterizes a constant function, making it an unchanging horizontal line when graphed.

Function Analysis: Understanding the Behavior of Functions

Function analysis involves studying the properties and characteristics of functions, such as their continuity, intervals of increase or decrease, and limits. In the context of the given exercise, proper function analysis leads us to conclude that the function is constant.

• The consistency of \( f(x) \) stems from analyzing the derivative, \( f'(x) = 0 \), which suggests uniformity in function behavior over all \( x \) values.
• A constant function will have all of its outputs as the same, signifying no change across any input value.
• Graphically, this is represented as a straight horizontal line, which helps in visualizing the nature of constant functions.

When analyzing a function like this, understanding the implications of a zero derivative across the entire domain often simplifies broader function behavior analysis, affirming the function's constancy.

Initial Conditions and Their Role

When dealing with functions, initial conditions are pieces of information given about the value of the function at a specific point. These conditions are critical as they help in defining or determining the function's behavior.
In our exercise, we know \( f(-1) = 3 \). Coupled with the knowledge that \( f(x) \) is constant due to \( f'(x) = 0 \), we utilize this initial condition to precisely identify the constant value of the entire function.

• An initial condition like \( f(-1) = 3 \) allows us to assign the constant value for \( f(x) \) everywhere on its domain since the derivative is zero.
• This practical use simplifies the process of finding the constant value of \( f(x) \), making initial conditions essential in work involving constant or differential equations.

Therefore, understanding and applying initial conditions effectively allows us to solve for specific constants that determine the function's entire output range.