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Chapter 3: Problem 4

Determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=\frac{x^{2}-1}{2 x+1} $$

Short Answer

Expert verified
The function is concave upward in the intervals \( (-\infty, -1) \) and \( (-1/2, 0) \). It is concave downward in the interval \( (0, +\infty) \).

Step by step solution

01

Differentiate the function

Differentiate \( f(x) = \frac{x^{2}-1}{2x+1} \) using the quotient rule for differentiation, which gives us \( f'(x) = \frac{(2x)(2x+1)-(x^{2}-1)(2)}{(2x+1)^{2}} \). Simplify \( f'(x) \) to get \( \frac{2x^{2}+2x-x^{2}+2}{4x^{2}+4x+1} \) which simplifies further to \( \frac{x^{2}+2}{4x^{2}+4x+1} \).
02

Differentiate \( f'(x) \) to get the second derivative

Differentiate the first derivative \( f'(x) = \frac{x^{2}+2}{4x^{2}+4x+1} \) using the quotient rule to get the second derivative \( f''(x) \). Thus, \( f''(x) = \frac{2x(4x^{2}+4x+1)-(x^{2}+2)(8x+4)}{(4x^{2}+4x+1)^{2}} \). Simplify to get \( f''(x) = \frac{-2x^{2}-12x-4}{(4x^{2}+4x+1)^{2}} \). Now, identify where \( f''(x) \) is positive and negative.
03

Solve \( f''(x) > 0 \) to find where the function is concave up

Set \( f''(x) > 0 \) and solve for \( x \). This gives \( \frac{-2x^{2}-12x-4}{(4x^{2}+4x+1)^{2}} > 0 \), which gives \( x < -1, -1/2 \). The function is therefore concave upward in the intervals \( (- \infty, -1) \) and \((-1/2, 0) \).
04

Solve \( f''(x) < 0 \) to find where the function is concave down

Set \( f''(x) < 0 \) and solve for \( x \). This gives \( \frac{-2x^{2}-12x-4}{(4x^{2}+4x+1)^{2}} < 0 \), which gives \( x > 0 \). Therefore, the function is concave downward in the interval \( (0, +\infty) \)

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Key Concepts

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

Quotient Rule
When dealing with functions that are expressed as a ratio of two functions, the quotient rule is a handy differentiation tool. It's a method used to find the derivative of a function that is the quotient of two differentiable functions. The general formula is:
  • If you have a function given by \( f(x) = \frac{g(x)}{h(x)} \), then its derivative is: \[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]
This rule is crucial when you need to find the rate of change of a function in this form. First, differentiate the numerator and denominator separately, then apply the formula. In our exercise, this rule helped in calculating both the first and second derivatives of the function \( f(x) = \frac{x^{2}-1}{2x+1} \). Each time, the differentiation involved simplifying the results to make further calculations easier.
Second Derivative Test
This test is useful for analyzing the concavity of a function's graph and identifying points of inflection. After computing the second derivative of a function, you can determine the concavity by checking whether \( f''(x) \) is positive or negative.
  • If \( f''(x) > 0 \), the graph is concave up.
  • If \( f''(x) < 0 \), the graph is concave down.
In the given exercise, we used the second derivative test to find where \( f(x) = \frac{x^{2}-1}{2x+1} \) was concave upward or downward. By setting \( f''(x) \) greater or less than zero, we could divide the domain into intervals of different concavity characteristics. This helps in understanding the behavior of the function graphically and deciding where it is bending upwards or downwards.
Differentiation
Differentiation, the process of finding a derivative, is a fundamental tool in calculus used to understand how a function changes. A derivative represents the function鈥檚 rate of change or the slope of the tangent line at any given point on its graph.
  • The first derivative provides information about the increase or decrease of the function. This is done using various rules like product, quotient, and chain rule.
  • The second derivative offers insights into the curve's concavity and potential inflection points.
In our exercise, we differentiated the function \( f(x) \) twice. The first derivative was obtained using the quotient rule to understand the initial rate of change. Then, the second differentiation provided insights into the concavity, confirming whether the sections of the curve were concave up or down.
Open Intervals
Open intervals are essential for describing the domain of a function where certain properties hold. An open interval is essentially a range of x-values that does not include its endpoints, denoted as \((a, b)\).
  • Analyzing open intervals can tell us where a function is behaving in a specific way, such as increasing, decreasing, concave up, or concave down.
In calculus problems like the one from our exercise, open intervals help define sections where the function maintains certain characteristics. For instance, after identifying where the second derivative of \( f(x) \) is positive or negative, we found intervals such as \((-\infty, -1)\) and \((0, +\infty)\) to indicate where the graph is concave up or down, respectively. This concept aids in interpreting the behavior of graphed functions efficiently.

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