Problem 29 - University Calculus: Early Transcendentals

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 the derivative is zero, indicating a constant function.

Step 1: Understanding the Given Information

We are given that the value of the function at \( x = -1 \) is 3, i.e., \( f(-1) = 3 \). Furthermore, the derivative of the function \( f'(x) = 0 \) for all \( x \). The derivative being zero suggests something specific about the nature of \( f(x) \).

Step 2: Interpret the Derivative Condition

Since \( f'(x) = 0 \) for all \( x \), this implies that \( f(x) \) has a constant slope everywhere. A zero slope means the function does not change and is constant for all values of \( x \).

Step 3: Determine the Form of the Function

A function with a derivative of zero everywhere is a constant function. Therefore, \( f(x) \) must be a constant function, i.e., \( f(x) = c \) for some constant \( c \).

Step 4: Set Constant Value Using Given Point

Since we know that \( f(-1) = 3 \) and we have established that \( f(x) \) is constant, it follows that \( c = 3 \). Thus, the function \( f(x) = 3 \) for all \( x \).

Key Concepts

Understanding Derivatives

A derivative is a fundamental concept in calculus that tells us how a function changes at any given point.
It gives the slope of the tangent line to the curve of the function at a specific point.

• If the derivative of a function at a certain point is positive, the function is increasing at that point.
• If it's negative, the function is decreasing.
• If the derivative is zero, the function might be at a peak, valley, or constant over that region.

In the exercise, it mentions that the derivative, \( f'(x) \), is zero for all \( x \).
This leads us to understand that at every point, the rate of change or slope is zero, indicating that the function does not rise or fall.Thus, the function must not change as \(x\) changes, revealing that it is a constant function.

Implications of a Zero Slope

When we talk about slope, we usually refer to how steep a line is, which is directly connected to the derivative in calculus.
If the slope (or derivative) of a function is zero, the line is perfectly horizontal.Here's why zero slope is crucial:

• A zero slope indicates that no matter which two points you choose on the function, the function value remains the same between them.
• This is because there is no rise or fall in the values - a hallmark of a constant function.

This concept helps reinforce the idea in the exercise.
If a function's derivative is zero everywhere, it confirms that the entire function is flat and constant. Therefore, even though the exercise specifies \( f(-1) = 3 \), the constant nature implies \( f(x) = 3 \) for all \( x \).

Recognizing Constant Value

A constant value in a function means the function outputs the same value regardless of the input variable.
This can easily be linked to the concept of having a zero derivative.Consider these characteristics:

• The function does not vary; hence, there are no ups or downs as \( x \) moves.
• The graph of a constant function is a straight, horizontal line.

In the given exercise, we're told that \( f(-1) = 3 \), and the derivative is zero throughout. Using this point of reference, alongside the understanding of what a zero derivative means, it leads us to conclude that \( f(x) = 3 \) universally.Every \( x \) value inputted into the function results in the same constant output of 3.