# Activity9.3Formal Identification of Critical Numbers¶ permalink

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?