Problem 39 - Calculus with Concepts in Calculus

Question

Suppose \(c\) and \(d\) are not both 0 , and let $$ f(x)=\frac{a x+b}{c x+d} $$ Show that \(f\) has no critical numbers unless \(a d-b c=0\), in which case \(f\) is a constant function.

Step-by-Step Solution

Answer

The function has no critical numbers unless \(ad - bc = 0\), which makes \(f(x)\) constant.

Step 1: Understanding the Problem

We are given the function \(f(x) = \frac{a x + b}{c x + d}\). We need to determine the conditions under which this function has no critical numbers, and in particular, show that this occurs unless \(ad - bc = 0\).

Step 2: Finding the Derivative

To find critical numbers, we need the derivative of \(f(x)\). Using the quotient rule, \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\), where \(u(x) = ax + b\) and \(v(x) = cx + d\).

Step 3: Calculating Derivative Components

Calculate the derivatives: \(u'(x) = a\) and \(v'(x) = c\). According to the quotient rule, the derivative \(f'(x)\) becomes:\[ f'(x) = \frac{a(cx + d) - c(ax + b)}{(cx + d)^2} \].

Step 4: Simplifying the Derivative

Simplifying the numerator: \[ a(cx + d) - c(ax + b) = acx + ad - acx - bc = ad - bc \]. Hence, the derivative is: \[ f'(x) = \frac{ad - bc}{(cx + d)^2} \].

Step 5: Analyzing Critical Numbers

Critical numbers occur where \(f'(x) = 0\) or where \(f'(x)\) is undefined. For \(f'(x) = 0\), the numerator must be 0, i.e., \(ad - bc = 0\). The denominator \((cx + d)^2\) is never 0 since \(c\) and \(d\) are not both zero.

Step 6: Conclusion on Critical Numbers

Since the only way for \(f'(x) = 0\) is if \(ad - bc = 0\), which makes \(f(x)\) a constant (since \(f'(x) = 0\) for all \(x\)), otherwise \(f(x)\) has no critical numbers.

Step 7: Verifying Constant Function Condition

If \(ad - bc = 0\), then \(f'(x) = 0\), meaning \(f(x)\) does not change, confirming it is a constant function. Thus, the condition \(ad - bc = 0\) indeed ensures \(f(x)\) is constant.

Key Concepts

Derivative of Rational Functions

The derivative of rational functions is an essential concept in calculus, particularly when dealing with functions that can be expressed as the quotient of two polynomials. A rational function is of the form \( f(x) = \frac{N(x)}{D(x)} \), where both \(N(x)\) and \(D(x)\) are polynomials. Calculating the derivative of such functions often requires using specific rules of differentiation, as they typically involve both a numerator and a denominator function.

To find the derivative of a rational function, it's essential to manage both the change in the numerator and the change in the denominator. This is achieved by using rules such as the Quotient Rule, which will be discussed in the next section. Derivatives provide insights into the behavior of functions, such as increasing or decreasing intervals, and identifying critical points where the slope is zero, signaling potential local maxima, minima, or points of inflection.

Quotient Rule

The Quotient Rule is a fundamental tool in calculus for finding the derivative of a quotient of two functions. When you have a function defined as a division, like \( f(x) = \frac{u(x)}{v(x)} \), the quotient rule is used to compute its derivative. This rule is crucial for handling rational functions where both the numerator and the denominator are subject to change.

The Quotient Rule is given by the formula:

• \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

Here, \(u'\) and \(v'\) represent the derivatives of \(u(x)\) and \(v(x)\), respectively. The rule essentially combines the changes in function components, ensuring that the overall derivative respects both parts of the quotient.

Understanding this rule allows you to tackle complex rational functions by correctly balancing the effects of changing both the numerator and the denominator. This balance is vital for applications in both applied mathematics and physics settings, where these functions often occur.

Constant Functions

A constant function is one of the simplest types of functions, where the output value remains the same regardless of the input. Mathematically, a constant function can be expressed as \( f(x) = c \), where \(c\) is a constant. This implies that the derivative of a constant function is zero, \( f'(x) = 0 \).

In the context of the problem given, calculating \( f'(x) = 0 \) results in the insight that \( f(x) \) must be constant if the numerator of the rational function's derivative equals zero. Specifically, if \( ad - bc = 0 \), this implies that the function derived is a constant, regardless of \(x\).

Identifying constant functions within the realm of rational functions is crucial because it simplifies the analysis of the function's behavior, confirming that no critical points exist since the function doesn't increase or decrease.