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Activity9.3Formal Identification of Critical Numbers

When finding critical numbers based upon a function formula, there are three issues that need to be considered; the domain of the function, the zeros of the first derivative, and the numbers in the domain of the function where the first derivative is undefined. When writing a formal analysis of this process each of these questions must be explicitly addressed. The following outline shows the work you need to show when you are asked to write a formal determination of critical numbers based upon a function formula.

Subsection9.3.1Exercises

Formally establish the critical numbers for each of the following functions following the procedure outlined in Algorithm 9.3.1.

1

\(\fe{f}{x}=x^2-9x+4\)

2

\(\fe{g}{t}=7t^3+39t^2-24t\)

3

\(\fe{p}{t}=(t+8)^{\sfrac{2}{3}}\)

4

\(\fe{z}{x}=x\fe{\ln}{x}\)

5

\(\fe{y}{\theta}=e^{\fe{\cos}{\theta}}\)

6

\(\fe{T}{t}=\sqrt{t-4}\sqrt{16-t}\)

The first derivative of the function \(\fe{m}{x}=\frac{\sqrt{x-5}}{x-7}\) is \begin{equation*}\fe{\fd{m}}{x}=\frac{3-x}{2\sqrt{x-5}(x-7)^2}\text{.}\end{equation*}

7

Roland says that \(5\) is a critical number of \(m\) but Yuna disagrees. Who is correct and why?

8

Roland says that \(7\) is a critical number of \(m\) but Yuna disagrees. Who is correct and why?

9

Roland says that \(3\) is a critical number of \(m\) but Yuna disagrees. Who is correct and why?