Problem 5 - Calculus Early Transcendentals

Question

Determine whether the function \(f\) is one-to-one by examining the sign of \(f^{\prime}(x)\). $$ \begin{array}{l}{\text { (a) } f(x)=x^{2}+8 x+1} \\ {\text { (b) } f(x)=2 x^{5}+x^{3}+3 x+2} \\ {\text { (c) } f(x)=2 x+\sin x} \\ {\text { (d) } f(x)=\left(\frac{1}{2}\right)^{x}}\end{array} $$

Step-by-Step Solution

Answer

Functions (b), (c), and (d) are one-to-one, while (a) is not.

Step 1: Differentiate function (a)

Compute the derivative of the function \(f(x) = x^2 + 8x + 1\):\[f'(x) = 2x + 8\] Analyze the sign of \(f'(x)\). The derivative is linear, and \(f'(x) = 2x + 8 > 0\) for \(x > -4\) and \(f'(x) < 0\) for \(x < -4\). Hence, the function is not one-to-one on the real number domain.

Step 2: Differentiate function (b)

Compute the derivative of the function \(f(x) = 2x^5 + x^3 + 3x + 2\):\[f'(x) = 10x^4 + 3x^2 + 3\] Since all the coefficients of the polynomial are non-negative and all powers are even or zero, \(f'(x)\) is always positive for any \(x\). Therefore, the function is one-to-one.

Step 3: Differentiate function (c)

Compute the derivative of the function \(f(x) = 2x + \sin x\):\[f'(x) = 2 + \cos x\] Since \(-1 \leq \cos x \leq 1\), the minimum possible value of \(f'(x)\) is \(2 - 1 = 1\). Therefore, \(f'(x)\) is positive for all \(x\), indicating that the function is one-to-one.

Step 4: Differentiate function (d)

For the function \(f(x) = \left(\frac{1}{2}\right)^x\), compute the derivative:\[f'(x) = \ln\left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right)^x\] Since \(\ln\left(\frac{1}{2}\right)\) is negative, \(f'(x) < 0\) for all \(x\). The function is decreasing everywhere on its domain, indicating that it is one-to-one.

Key Concepts

Derivative Test

When examining whether a function is one-to-one, the Derivative Test is a powerful tool. This test involves taking the derivative of a function and analyzing its sign to determine the function's nature.
The idea is simple:

• If the derivative, \( f'(x) \), is positive over an interval, the function is increasing in that interval.
• If \( f'(x) \) is negative, the function decreases.

For a function to be one-to-one, it must consistently increase or consistently decrease throughout its domain. That means the derivative should not change sign.

Applying the Derivative Test

Consider the function \( f(x) = x^2 + 8x + 1 \). The derivative is \( f'(x) = 2x + 8 \). By analyzing the derivative:

• If \( x > -4 \), then \( f'(x) > 0 \): the function increases.
• If \( x < -4 \), then \( f'(x) < 0 \): the function decreases.

Here, the derivative changes sign, indicating that the function is not one-to-one on the real domain. This method provides a straightforward way to identify whether a function can be one-to-one.

Function Analysis

Function Analysis encompasses evaluating a function's growth or decline using its derivative. This helps in understanding behavioral patterns of functions.
Each derivative tells us about the function's slope at any given point.

• For polynomial functions, derivatives can reflect constant sign throughout the domain.
• For trigonometric functions, the periodic nature of derivatives influences the function's behavior.
• Exponential derivatives are characterized by their rate of growth or decay.

Analyzing Specific Functions

Take \( f(x) = 2x^5 + x^3 + 3x + 2 \) as an example. Here, \( f'(x) = 10x^4 + 3x^2 + 3 \). Given the nature of its terms:

• Even powers in \( f'(x) \) result in all positive values, denoting upward growth.
• No change in sign over any interval ensures the function remains one-to-one.

This analysis helps to verify whether the varied nature of derivatives impacts a function's one-to-one characteristic.

Calculus Problem Solving

Problem-solving in calculus often draws heavily from understanding derivatives and their implications. It’s a process of translating derivative-based information into real-world behavior of functions.

Steps for Solving Problems

• Differentiation: Compute the derivative of a given function.
• Sign Analysis: Investigate the sign (positive or negative) of the derivative for various intervals to predict function behavior.
• Conclusion: Use these signs to conclude if the function is consistently increasing or decreasing.

For example, consider \( f(x) = 2x + \sin x \). The derivative, \( f'(x) = 2 + \cos x \), guarantees positivity as it lies between 1 and 3. Hence, indicating a strictly increasing trend, confirming its one-to-one nature.
Each problem becomes tractable when results from differentiation guide us in understanding the broader behavior of functions.