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Chapter 1: Problem 36

Describing Function Behavior Determine the intervals on which the function is increasing, decreasing, or constant. $$f(x)=\frac{x^{2}+x+1}{x+1}$$

Short Answer

Expert verified
The function \(f(x) = (x^2 + x + 1) / (x + 1) \) decreases on the interval (- \infty, -1), increases on the interval (-1, 0) and decreases again on the interval (0, \infty). The function has no intervals where it remains constant.

Step by step solution

01

Differentiate the function

To find where the function increases or decreases, we need to find its derivative. The derivative of the given function can be calculated using the quotient rule, which states: If we have a function in the form \(h(x) = g(x) / f(x)\), then its derivative is given by \(h'(x) = (g'(x)f(x) - g(x)f'(x)) / (f(x))^2\). Applying this rule to our function: \(f'(x) = (2x+1)(x+1) - (x^{2}+x+1) / (x+1)^2\). Simplifying this gives \(f'(x) = x / (x+1)^2\).
02

Find Critical Points

Critical points occur at values of \(x\) where the derivative is zero or undefined. Set the numerator of \(f'(x)\) to zero and find the values of \(x\), which gives \(x = 0\). f'(x) is undefined when the denominator equals zero. Equating \((x+1)^2 = 0\), we find \(x = -1\). These are the critical points of the function \(x = 0\) and \(x = -1\).
03

Determine Increasing or Decreasing Intervals

The function increases on intervals where \(f'(x) > 0\) and decreases where \(f'(x) < 0\). Taking the domain of the given function into account (- \infty, -1), (-1, 0), and (0, \infty), test each interval with a value into \(f'(x)\) to ascertain if it's positive or negative. If we choose \(x = -2\), \(x = -0.5\), and \(x = 1\), we obtain \(f'(-2) < 0\), \(f'(-0.5) > 0\) and \(f'(1) < 0\). Therefore, the function decreases on (-\infty, -1), increases on (-1, 0), and decreases again on (0, \infty).
04

Identify Constant Intervals

A function is constant at a point where the derivative is equal to zero. However, for it to be constant over an interval, the derivative must equal zero throughout that interval which is not the case in this function. Thus, the function has no intervals where it is constant.

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Key Concepts

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

Derivative
Understanding the derivative is crucial for analyzing the behavior of a function. The derivative is a mathematical tool that measures how a function changes as its input changes.
It tells us the rate at which the function's value is increasing or decreasing. In simple terms, it gives us a slope of the tangent line at any point on the function's graph.
In the context of our exercise, to find the derivative of the function \( f(x)=\frac{x^{2}+x+1}{x+1} \), we use the quotient rule. This rule is essential when calculating the derivative of a ratio of two functions.
  • The quotient rule formula is \( h'(x) = \frac{g'(x)f(x) - g(x)f'(x)}{(f(x))^2} \).
  • Applying the quotient rule to our function helps us understand where the function is changing and at what rate.
  • After applying the rule, we find \( f'(x) = \frac{x}{(x+1)^2} \).
  • This derivative will be key in determining where the function is increasing or decreasing.
Increasing and Decreasing Intervals
With the derivative in hand, we can now find the increasing and decreasing intervals of the function. These intervals tell us where the function is rising and where it is falling.
The sign of the derivative indicates these behaviors:
  • If \( f'(x) > 0 \), the function is increasing on that interval.
  • If \( f'(x) < 0 \), the function is decreasing on that interval.
For the function \( f(x)=\frac{x^{2}+x+1}{x+1} \), our derivative \( f'(x) = \frac{x}{(x+1)^2} \) changes sign across different intervals:
  • When testing the interval \((-\infty, -1)\) with a value such as \(x = -2\), the derivative is negative, indicating the function is decreasing.
  • For the interval \((-1, 0)\) using \(x = -0.5\), the derivative is positive, showing the function is increasing.
  • In the interval \((0, \infty)\), using \(x = 1\), we again find the derivative is negative, meaning the function decreases once more.
Critical Points
Critical points are specific values of \( x \) at which a function's derivative is either zero or undefined. These points are significant because they often indicate where important changes in the function occur.
In our exercise:
  • The derivative \( f'(x) = \frac{x}{(x+1)^2} \) is equal to zero when \( x = 0 \). This is one critical point because the slope of the tangent line here is horizontal.
  • The derivative becomes undefined when the denominator is zero. Here, this occurs when \( x = -1 \). This point is also crucial because it suggests a potential vertical tangent or a discontinuity.
Identifying these critical points helps us to decide which intervals we need to test for increasing or decreasing behavior. However, it's also essential to note whether these points are actually part of the domain, given the context of the original function.

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