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Chapter 1: Problem 44

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

Short Answer

Expert verified
The function is decreasing on the interval \( x < -1 \), increasing on the interval \( -1 < x < 0 \), and increasing on the interval \( 0 < x \). The function has a point of discontinuity at \( x = -1 \).

Step by step solution

01

Find the Derivative

Differentiation of the function \( f(x) = \frac{x^{2} + x + 1}{x+1} \) is the first step. Use the quotient rule to do this: \( f'(x) = \frac{(x+1) (2x+1) - (x^{2} + x + 1)}{(x+1)^{2}} \). Simplifying this, we have \(f'(x) = \frac{x}{(x+1)^2}\).
02

Determine Critical Values

Critical values occur where the derivative is either zero or undefined. The derivative, \( f'(x) = \frac{x}{(x+1)^2} \), is zero when \( x = 0 \). It is undefined when \( x=-1 \). Here, \( -1 \) also makes the function undefined, so it is a point of discontinuity.
03

Use the First Derivative Test

To find out whether the function is increasing or decreasing at different intervals, we create a sign chart. Consider the values of \( x < -1 \), \( -1 < x < 0 \), and \( 0 < x \). Plug these ranges into the first derivative function, \( f'(x) = \frac{x}{(x+1)^2} \). The signs of results (positive or negative) tell us where the function is increasing or decreasing, respectively.
04

Interpret the Results

The function is decreasing on the interval \( x < -1 \), increasing on the interval \( -1 < x < 0 \), and increasing on the interval \( 0 < x \). The critical point \( x = 0 \) is a local minimum as the function changes from decreasing to increasing at this point. The function has a point of discontinuity at \( x = -1 \). There is no interval where the function is constant as the derivative is never zero.

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Most popular questions from this chapter

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=\frac{1}{4} $$

Determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$

Find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(y\) is inversely proportional to \(x .(y=7\) when \(x=4\).)

Find a mathematical model for the verbal statement. The rate of growth \(R\) of a population is jointly proportional to the size \(S\) of the population and the difference between \(S\) and the maximum population size \(L\) that the environment can support.

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{6 x+4}{4 x+5} $$
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