Chapter 1.3, Problem 38E

(a)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

The function $f(x)=x\sqrt{3-x}$ .

Graph:

The given function is sketched as shown below.

  

Interpretation

From the graph, it can be seen that the given function is initially increasing in nature over a range of values and then decreases with increase in x .

(b)

To determine

To find: the nature of the function from the graph and using a table of values of the function.

Answer to Problem 38E

The given function increases in the range $(-\infty ,2)$ and then decreases in the range $(2,3)$ .

Explanation of Solution

Given information:

The function $f(x)=x\sqrt{3-x}$ .

From the graph of the function in previous part, it can be seen that, for all values of x from -8 to 2, the value of f (x ) increases. Then, in the range 2 to 3, it decreases. Thus, $f(x)=x\sqrt{3-x}$ is a increasing function in the range $(-\infty ,2)$ and decreasing function in the range (2, 3).

Now, the values of the given function for different values of x are shown in the below table.

  

From the table, it can be verified that, for all the values of x considered, the function initially shows a increasing nature and the root point being x =2 from where, the function shows a steady decrease to reach 0 at x =3.