Problem 23 - Thomas Calculus

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, because the derivative is zero, indicating a constant function, and \(f(-1) = 3\).

Step 1: Understanding the Given Information

We are given two key pieces of information: 1) The function value at \(x = -1\) is \(f(-1) = 3\). 2) The derivative of the function \(f\), denoted \(f'(x)\), is 0 for all \(x\). When a function's derivative is zero everywhere, it implies the function is constant.

Step 2: Analyzing Constant Derivative

Since \(f'(x) = 0\) for all \(x\), this means that the slope of the tangent to the graph of \(f\) at any point is zero. A zero slope indicates a horizontal line, therefore a constant function.

Step 3: Concluding the Function's Form

A function with a zero derivative everywhere must be constant across its entire domain. Given the point \(f(-1) = 3\), if \(f\) is constant, \(f(x)\) must equal 3 for all \(x\).

Step 4: Final Answer with Reasoning

Since \(f'(x) = 0\) implies \(f\) is constant, and we know \(f(-1) = 3\), it follows that \(f(x) = 3\) for all \(x\). Hence, the assertion "\(f(x) = 3\) for all \(x\)" is true.

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that describes how a function changes as its input changes. It's an essential tool for understanding the behavior of functions. The derivative at a point can be thought of as the slope of the tangent line to the curve at that point.
When we say that the derivative, denoted as \( f'(x) \), is zero for all \( x \), it means that the function does not change as \( x \) changes. This suggests that, regardless of where you are on the graph, the slope of the tangent is horizontal.

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.
• If \( f'(x) = 0 \), the function has a constant slope.

Thus, a zero derivative over the entire domain indicates a constant function.

Constant Function

A constant function is one of the simplest functions in calculus. As its name suggests, a constant function remains the same, regardless of the input value. For a function \( f(x) \) to be constant, its derivative throughout its domain must be zero, as derived from the concept of derivatives.
Constant functions are represented by horizontal lines on a graph. This arises from the fact that a constant slope of zero implies no change in the value of \( f(x) \) as \( x \) changes.

• Example: If \( f(x) = c \) for all \( x \), where \( c \) is a constant, the graph is a horizontal line at \( y = c \).
• Real-world analogy: Walking on a flat treadmill where your height remains constant.

In the context of our exercise, because the function \( f(x) \) has a zero derivative and \( f(-1) = 3 \), we conclude \( f(x) = 3 \) for all \( x \).

Function Analysis

Function analysis involves carefully examining different properties and behaviors of a given function. By analyzing a function, you can understand its tendency to increase, decrease, or remain constant.
Let's explore the basic steps involved in a typical function analysis:

• Determine the domain and range: Understand where the function is defined and the possible values it can take.
• Identify critical points: Use derivatives to find where the function has maxima, minima, or points of inflection.
• Consider the overall behavior: Look at whether the function is constant, increasing, or decreasing across its entire domain.

In our exercise, function analysis helps verify that \( f'(x) = 0 \) indeed means the function is constant. Since we know one value of the function, \( f(-1) = 3 \), and we know the function is constant, we conclude \( f(x) = 3 \) for all \( x \). This conclusion relies on understanding how derivatives reveal the constancy or variability of functions.