Chapter 3.2, Problem 99E

a.

To determine

To graph:

The given function using a graphing utility.

Answer to Problem 99E

The graph of our given function would be:

  

Explanation of Solution

Given:

A function: $f(x)=\frac{x}{2}-\ln \left(\frac{x}{4}\right)$

Calculation:

Upon graphing the given functionusing a graphing utility, we will get our required graph as shown below:

  

b.

To determine

To find:

The domain of the given function.

Answer to Problem 99E

The domain of our given logarithmic function would be $x>0$ or $(0,\infty )$ .

Explanation of Solution

Given:

A function: $f(x)=\frac{x}{2}-\ln \left(\frac{x}{4}\right)$

Calculation:

We know that logarithmic functions are not defined for negative values. To find domain of our given function, we will set argument of log function greater than 0 as:

  $\begin{array}{l}\frac{x}{4}>0\\ \frac{x}{4}\cdot 4>0\cdot 4\\ x>0\end{array}$

Therefore, the domain of our given logarithmic function would be $x>0$ or $(0,\infty )$ .

c.

To determine

To find:

Open intervals on which the given function is increasing and decreasing using the graph of function.

Answer to Problem 99E

The function is decreasing on interval $(0,2)$ and increasing on $(2,\infty )$ .

Explanation of Solution

Given:

A function: $f(x)=\frac{x}{2}-\ln \left(\frac{x}{4}\right)$

Calculation:

Upon looking at graph of our given function, we can see that function is decreasing from 0 to 2 and increasing from 2 to positive infinity.

Therefore, the function is decreasing on interval $(0,2)$ and increasing on $(2,\infty )$.

d.

To determine

To approximate:

The relative maximum or minimum values of the function. Round to three decimal places.

Answer to Problem 99E

The relative minimum of the given function is $(2,1.693)$ .

Explanation of Solution

Given:

A function: $f(x)=\frac{x}{2}-\ln \left(\frac{x}{4}\right)$

Calculation:

We can see from our given graph that there is no maximum value for our given function.

We can see that the minimum value of our given function occurs at $x=2$ that is $1.693$.

Therefore, the relative minimum of the given function is $(2,1.693)$ .