Chapter 2.2, Problem 46E

(a)

To determine

To find: all the real zeros of the given function.

Answer to Problem 46E

The real zeros are $x=0,\pm \sqrt{2}$

Explanation of Solution

Given information:

Consider $f(x)=x^5+x^3-6x$

Calculation:

Find the zero’s of the given function

The function

  $\begin{array}{l}f(x)=x^5+x^3-6x=0\\ x(x^4+x^2-6)=0\\ x=0\\ x^4+x^2-6=0\\ $ \Rightarrow {(x^2+3)}^2(x^2-2)=0$x^2=-3,2\\ x=\pm \sqrt{2},\pm i\sqrt{3}\end{array}$

Hence, the real zeros are $x=0,\pm \sqrt{2}$

(b)

To determine

To find: whether the multiplicity of the each zero of the given function is even or odd .

Answer to Problem 46E

Multiplicity of $x=0$ is odd and

Multiplicity of $x=\pm \sqrt{2}$ is even.

Explanation of Solution

Given information:

Consider $f(x)=x^5+x^3-6x$

Calculation:

From part a,

The real zeros are $x=0,\pm \sqrt{2}$

Multiplicity of $x=0$ is $=1$

Multiplicity of $x=\pm \sqrt{2}$ is $=2$

Hence, Multiplicity of $x=0$ is odd and

Multiplicity of $x=\pm \sqrt{2}$ is even

(c)

To determine

To find: the maximum possible number of turning points of the given function.

Answer to Problem 46E

The maximum number of turning points $=4$

Explanation of Solution

Given information:

Consider $f(x)=x^5+x^3-6x$

Calculation:

The maximum number of turning points is one less than the degree of the function

Degree of the function is $=5$

Hence, the maximum number of turning points $=4$

(d)

To determine

To plot: the given function and to verify the results of previous parts.

Explanation of Solution

Given information:

Consider $f(x)=x^5+x^3-6x$

Graph:

The graph of the $f(x)=x^5+x^3-6x$ is shown below

  

From the above graph, the maximum number of turning points $=2$

Hence, the results of previous parts are correct from the graph except maximum number

of turning points.