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Chapter 4: Problem 11

Describe the concavity of the graph and find the points of inflection (if any). $$f(x)=\frac{x}{x^{2}-1}$$.

Short Answer

Expert verified
The concavity of the graph and the inflection points are determined by the second derivative of the function. By solving the second derivative to zero, the x-values for inflection points can be obtained, and substituting them back into the original function will produce the corresponding y-values. The intervals over which the function is concave upward or downward can also be calculated from the analysis of the second derivative.

Step by step solution

01

Finding the Derivative

The first step is to calculate the first derivative of the function \(f(x)=\frac{x}{x^{2}-1}\). This computation involves applying the quotient rule for differentiation, which states that if you have a function of the form \(\frac{f}{g}\), its derivative is given by \(\frac{f'g - fg'}{g^{2}}\). So, the first derivative, \(f'(x)\), of the given function will be calculated.
02

Simplifying the First Derivative

After calculating, simplify the obtained expression for \(f'(x)\) by canceling common factors, combining like terms, etc.
03

Finding the Second Derivative

Now, find the second derivative of the function \(f(x)\). This will again involve application of the quotient rule for differentiation, just like in step 1.
04

Simplifying the second derivative

Now simplify the obtained expression for the second derivative, \(f''(x)\), in the same way as explained in step 2.
05

Equaling the Second Derivative to Zero

In this step, the simplified second derivative \(f''(x)\) is set to zero. This is because the points of inflection, where the graph changes its concavity, will yield zero at second differentiation.
06

Solve for x

Now, solve the equation obtained in step 5 for \(x\). The solutions gotten are the x-coordinates of the inflection points.
07

Substitute in the Original Function

Lastly, substitute the obtained x-values into the original function \(f(x)\) to get the corresponding y-values, thus getting the points of inflection.
08

Analyzing the Second Derivative

The concavity of the function on different intervals can further deduced by analyzing the sign of the second derivative \(f''(x)\) in those intervals. Defines areas where \(f''(x) > 0\) as concave upward, and where \(f''(x) < 0\) as concave downward.

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Key Concepts

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

Quotient Rule for Differentiation
Differentiating complex functions might seem daunting, but the quotient rule for differentiation equips us with a systematic approach for functions presented as fractions. When faced with a function where one part is divided by another, this rule is essential. Simply put, if you have a function \( f(x) = \frac{u(x)}{v(x)} \), the quotient rule allows you to find its derivative, \( f'(x) \), using the formula \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \).

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Most popular questions from this chapter

An object that weighs 150 pounds on the surface of the earth will weigh \(150\left(1+\frac{1}{4000} r\right)^{-2}\) pounds when it is \(r\) miles above the earth. Given that the altitude of the object is increasing at the rate of 10 miles per minute, how fast is the weight decreasing which the object is 400 miles above the surface?

A rope 32 feet long is attached to a weight and passed over a pulley 16 feet above the ground. The other end of the rope is pulled away along the ground at the rate of 3 feet per second. At what rate is the angle between the rope and the ground changing when the weight is exactly 4 feet off the ground?

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