Problem 66 - Essential Calculus Early Transcendentals

Question

For what values of \(c\) is the function $$f(x)=c x+\frac{1}{x^{2}+3}$$ increasing on \((-\infty, \infty) ?\)

Step-by-Step Solution

Answer

The function is increasing for all \( c > 0 \).

Step 1: Understand the given function

The function represented is \( f(x) = c x + \frac{1}{x^2 + 3} \). We need to find the values of \( c \) for which this function is increasing on the entire interval \((-fty, fty)\).

Step 2: Find the derivative of the function

To determine when the function is increasing, we need to examine the derivative, \( f'(x) \). This involves differentiating each term: - The derivative of \( c x \) is \( c \).- The derivative of \( \frac{1}{x^2 + 3} \) is \(-\frac{2x}{(x^2 + 3)^2} \) using the chain rule.

Step 3: Set up the inequality for increasing function

For \( f(x) \) to be increasing, the derivative \( f'(x) = c - \frac{2x}{(x^2 + 3)^2} \) must be positive for all \( x \). So, we set the inequality \( c - \frac{2x}{(x^2 + 3)^2} > 0 \).

Step 4: Analyze the inequality

Rearrange the inequality to: \[ c > \frac{2x}{(x^2 + 3)^2} \]Considering \( \frac{2x}{(x^2 + 3)^2} \), it is both an even and odd function, meaning it oscillates around 0. The maximum value happens when the numerator is maximized, but as you check at critical points or analyze algebraically, you see it stays below 1. Thus, \( c > 0\).

Step 5: Conclude for all x

The value of \( \frac{2x}{(x^2 + 3)^2} \) approaches 0 as \( x \to \infty \) and as \( x \to -\infty \), while for values \( x \to 0 \), it remains below 1. For \( f'(x) > 0 \), \( c \) must be positive.

Key Concepts

Increasing Functions

An increasing function is one where as you move along the x-axis, the y-values (or outputs) increase. This means that for any two points, say, \( x_1 \) and \( x_2 \) with \( x_1 < x_2 \), the relation \( f(x_1) < f(x_2) \) holds.

Understanding this is crucial when determining the behavior of functions over an interval. When a function is increasing, it indicates positive growth where the graph climbs upwards as you move from left to right.

• It's essential to note that some functions can increase over certain intervals and not others.
• Checking where a function is increasing involves using its derivative since the derivative indicates the rate of change of the function.

For our exercise, we need the function \( f(x) = c x + \frac{1}{x^2 + 3} \) to be increasing across the entire real number line. This requires that the derivative of \( f(x) \) be positive for all \( x \).

Derivatives

In calculus, a derivative is a tool that measures how a function changes as its input changes. This concept is key when analyzing whether a function like \( f(x) = c x + \frac{1}{x^2 + 3} \) is increasing or decreasing.

The derivative, symbolized as \( f'(x) \), gives the slope of the tangent line to the curve at any point \( x \). It tells how steep the curve is and whether it's going up or down at that specific point.

• A positive derivative indicates an increasing function; the slope of the tangent line is positive, showing an upward trend.
• A negative derivative suggests a decreasing function; the tangent's slope is negative, which means the function is going downwards.

For the given function, the derivative is \( f'(x) = c - \frac{2x}{(x^2 + 3)^2} \). We determine \( f(x) \) is increasing by finding when \( f'(x) > 0 \), leading to the condition \( c > \frac{2x}{(x^2 + 3)^2} \). Because we want \( f(x) \) to be increasing over all \( x \), we need \( c \) to be greater than the maximum value of the fractional term, found to be less than 1, hence \( c > 0 \).

Inequalities

Inequalities are mathematical expressions used to compare quantities. They are crucial when we need conditions like the function's derivative to be positive.

In this context, consider the inequality \( c - \frac{2x}{(x^2 + 3)^2} > 0 \). This inequality is derived from setting the derivative greater than zero to ensure the function is increasing.

• Rearranging inequalities often helps find solutions more clearly. Here, rearranging yields: \( c > \frac{2x}{(x^2 + 3)^2} \).
• Inequalities can sometimes oscillate, making it important to analyze their behavior, especially over all real numbers.

In our problem, the term \( \frac{2x}{(x^2 + 3)^2} \) has been analyzed to find its behavior. Its maximum does not exceed 1 for real \( x \), leading us to conclude \( c \) must be greater than 0 in order for the inequality to always hold true.