Chapter 1.3, Problem 34E

(a)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

The function $f(x)=3x^4-6x^2$ .

Graph:

The given function is sketched as shown below.

  

Interpretation

From the graph, it can be seen that the given function is initially decreasing in nature over a range of values and then increases with increase in x up to another point. Then, the function again decreases over a finite range of values of x and takes an increasing mode. It can also be observed that, the function is symmetric with respect to y -axis.

(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 34E

The given function decreases for value of x in the range $(-\infty ,-1)$ and then increases in the range $(-1,0)$ . Then it decreases in the range $(0,1)$ and finally increases in the range $(1,\infty )$ .

Explanation of Solution

Given information:

The function $f(x)=3x^4-6x^2$

From the graph of the function in previous part, it can be seen that, for all values of x from -8 to -1 , the value of f (x ) decreases. Then, in the range -1 to 0, it increases. Again, it decreases in the range (0,1) and then follows an increasing nature in the range (1, 8). Thus, $f(x)=3x^4-6x^2$ is a decreasing function in the range $(-\infty ,-1)$ and $(0,1)$ . It is increasing function in the range $(-1,0)$ and $(1,\infty )$ .

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, $f(x)=3x^4-6x^2$ is a decreasing function in the range $(-\infty ,-1)$ and $(0,1)$ . It is increasing in the range $(-1,0)$ and $(1,\infty )$ . The root points for the function is the values of x given by -1, 0,1. The symmetry of the function with respect to the y-axis can also be observed from the table of values.