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Activity9.5Inflection Points

When searching for inflection points on a function, you can narrow your search by identifying numbers where the function is continuous (from both directions) and the second derivative is either zero or undefined. (By definition an inflection point cannot occur at a number where the function is not continuous from both directions.) You can then build a sign table for the second derivative that implies the concavity of the given function.

When performing this analysis, you need to simplify the second derivative formula in the same way you simplify the first derivative formula when looking for critical numbers and local extreme points.

Subsection9.5.1Exercises

1

Identify the inflection points for the function shown in Figure 9.2.3.

The first two derivatives of the function \(\fe{y}{x}=\frac{(x+2)^2}{(x+3)^3}\) are \begin{equation*}\fe{\fd{y}}{x}=\frac{-x(x+2)}{(x+3)^4}\qquad\fe{\sd{y}}{x}=\frac{2(x+\sqrt{3})(x-\sqrt{3})}{(x+3)^5}\text{.}\end{equation*}

2

Yolanda was given this information and asked to find the inflection points on \(y\). The first thing Yolanda wrote was, “The critical numbers of \(y\) are \(\sqrt{3}\) and \(-\sqrt{3}\).” Explain to Yolanda why this is not true.

3

What are the critical numbers of \(y\) and in what way are they important when asked to identify the inflection points on \(y\)?

4

Copy Table 9.5.1 onto your paper and fill in the missing information.

Interval Sign of \(\sd{y}\) Behavior of \(y\)
\(\ointerval{-\infty}{-3}\)
\(\ointerval{-3}{-\sqrt{3}}\)
\(\ointerval{-\sqrt{3}}{\sqrt{3}}\)
\(\ointerval{\sqrt{3}}{\infty}\)
Table9.5.1\(\fe{\sd{y}}{x}=\frac{2\left(x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}{(x+3)^5}\)
5

State the inflection points on \(y\); you may round the dependent coordinate of each point to the nearest hundredth.

6

The function \(y\) has a vertical asymptote at \(-3\). Given that fact, it was impossible that \(y\) would have an inflection point at \(-3\). Why, then, did we never-the-less break the interval \(\ointerval{-\infty}{-\sqrt{3}}\) at \(-3\) when creating our concavity table?

Perform each of the following for the functions in Exercises 9.5.1.7–9.5.1.9.

  • State the domain of the function.

  • Find, and completely simplify, the formula for the second derivative of the function. It is not necessary to simplify the formula for the first derivative of the function.

  • State the values in the domain of the function where the second derivative is either zero or does not exist.

  • Create a table similar to Table 9.5.1. Number the table. Don't forget to include table headings and column headings.

  • State the inflection points on the function. Make sure that you explicitly address this question even if there are no inflection points.

7

\(\fe{f}{x}=x^4-12x^3+54x^2-10x+6\)

8

\(\fe{g}{x}=(x-2)^2e^x\)

9

\(\fe{G}{x}=\sqrt{x^3}+6\sqrt{x}\)

10

The second derivative of the function \(\fe{w}{t}=t^{1.5}-9t^{0.5}\) is \(\fe{\sd{w}}{t}=\frac{3(t+3)}{4t^{1.5}}\) yet \(w\) has no inflection points. Why is that?