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Chapter 4: Problem 12

Determine whether the function \(f\) is one-to-one by examining the sign of \(f^{\prime}(x)\). (a) \(f(x)=x^{3}+3 x^{2}-8\) (b) \(f(x)=x^{5}+8 x^{3}+2 x-1\) (c) \(f(x)=\frac{x}{x+1}\)

Short Answer

Expert verified
(a) Not one-to-one; (b) One-to-one; (c) One-to-one (except at the asymptote).

Step by step solution

01

Differentiate the Function (a)

The first step is to find the derivative of the function \( f(x) = x^3 + 3x^2 - 8 \).The derivative is: \[ f'(x) = 3x^2 + 6x \].
02

Analyze the Sign of the Derivative (a)

To determine if the function is one-to-one, we need to check if \( f'(x) \) is always positive or always negative.The derivative \( f'(x) = 3x^2 + 6x \) is a quadratic function and can be rewritten as \( 3x(x + 2) \).Check the sign of \( f'(x) \):- It is zero at \( x = 0 \) and \( x = -2 \).- For \( x < -2 \), \( f'(x) > 0 \).- For \( -2 < x < 0 \), \( f'(x) < 0 \).- For \( x > 0 \), \( f'(x) > 0 \).Since \( f'(x) \) changes sign, \( f(x) \) is not one-to-one.
03

Differentiate the Function (b)

Find the derivative of the function \( f(x) = x^5 + 8x^3 + 2x - 1 \).The derivative is: \[ f'(x) = 5x^4 + 24x^2 + 2 \].
04

Analyze the Sign of the Derivative (b)

Check if \( f'(x) = 5x^4 + 24x^2 + 2 \) is always positive or negative.Since all terms in \( f'(x) \) are squares or positive, the smallest it can be is when \( x = 0 \), yielding \( f'(0) = 2 \).Thus, \( f'(x) > 0 \) for all \( x \), and \( f(x) \) is one-to-one.
05

Differentiate the Function (c)

Find the derivative of \( f(x) = \frac{x}{x+1} \) using the quotient rule.\( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \).Here, \( u = x \) and \( v = x + 1 \), so \( u' = 1 \) and \( v' = 1 \).The derivative is: \[ f'(x) = \frac{(1)(x + 1) - (x)(1)}{(x + 1)^2} = \frac{1}{(x + 1)^2} \].
06

Analyze the Sign of the Derivative (c)

Since \( f'(x) = \frac{1}{(x + 1)^2} \) and the denominator \((x + 1)^2 \) is always positive, \( f'(x) > 0 \) for all \( x eq -1 \).Therefore, \( f(x) \) is one-to-one because \( f'(x) \) never changes sign (ignoring the vertical asymptote at \( x = -1 \)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative Analysis
Derivative analysis involves the study of changes represented by a function's derivative. In calculus, the derivative of a function gives us a lot of information about the original function's behavior such as its rate of change and direction. When examining whether a function is one-to-one, derivative analysis helps determine if the function is monotonically increasing or decreasing. For a function to be one-to-one, it must strictly increase or decrease over its entire domain.
  • If the derivative, denoted as \( f'(x) \), is positive for all \( x \), the function is increasing.
  • If \( f'(x) \) is negative, the function is decreasing.
  • If the derivative does not change sign, the function is monotonic and potentially one-to-one.
Understanding the derivative's behavior enables us to assess a function's injective property, which is crucial for determining if it is one-to-one.
Sign of the Derivative
The sign of the derivative tells us about the slope of the tangent line to the curve at any point along the graph of a function. By checking the sign of \( f'(x) \), we can understand whether the function is increasing or decreasing in different intervals. For \( f(x) = x^3 + 3x^2 - 8 \), the derivative \( f'(x) = 3x^2 + 6x \) was analyzed:
  • Where \( f'(x) > 0 \), the function is increasing.
  • Where \( f'(x) < 0 \), the function is decreasing.
A change in the sign of the derivative indicates a change in direction for the function's graph. For example, if \( f'(x) \) changes from positive to negative, the graph of the function changes from rising to falling, which often indicates a peak or turning point. If such a change in sign exists, the function is not strictly one-to-one.
Quadratic Functions
Quadratic functions frequently appear in calculus problems and derivative analysis. These functions are characterized by the form \( ax^2 + bx + c \). When examining their derivatives, you often find that:
  • The derivative is a linear function \( f'(x) = 2ax + b \), which is a straight line.
  • The sign of this derivative helps determine where the original quadratic is increasing or decreasing.
For instance, with a function like \( f(x) = x^3 + 3x^2 - 8 \), after differentiating, you receive a quadratic expression \( 3x(x + 2) \). By setting the derivative equal to zero, we find critical points, and the sign of intervals can be determined by testing values within those intervals. These characteristics are useful in determining when a function may fail to be one-to-one due to changes in direction.
Calculus Problems
Calculus problems often require analyzing functions for monotonicity and one-to-one properties, especially in determining when a function is injective. To solve such problems, you need to:
  • Find the derivative of the function.
  • Analyze the sign of the derivative to check behavior over intervals.
  • Look for zero points in the derivative, indicating changes in sign.
By applying these steps to each part of a calculus problem, as with the exercise with functions like \( x^3 + 3x^2 - 8 \), \( x^5 + 8x^3 + 2x - 1 \), and \( \frac{x}{x+1} \), you can determine injectivity. When the derivative is always positive or always negative, the function is one-to-one. However, if it changes sign, it may not be. This approach smoothly guides you through determining the nature of a function in various calculus contexts.

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