Chapter 1, Problem 14CT

(a)

To determine

To find: the zeros of the given function.

Answer to Problem 14CT

The zeros are at $x=-0.5,-1+\sqrt{2},-1-\sqrt{2}$

Explanation of Solution

Given information:

Consider function $f(x)=2x^6+5x^4-x^2$

Calculation:

Find the zeros

  $\begin{array}{l}f(x)=2x^6+5x^4-x^2=0\\ 2x^6+5x^4-x^2=0\\ x^2(2x^3+5x^2-1)=0\\ \Rightarrow x^2=0\\ x=0\\ \Rightarrow 2x^3+5x^2-1=0\\ (2x+1)(x^2+2x-1)=0\\ x=-0.5,-1+\sqrt{2},-1-\sqrt{2}\end{array}$

Hence, the zeros are at $x=-0.5,-1+\sqrt{2},-1-\sqrt{2}$

(b)

To determine

To sketch: the graph of the given function using graphing utility.

Explanation of Solution

Given information:

Consider function $f(x)=2x^6+5x^4-x^2$

Graph:

The graph of the $f(x)=2x^6+5x^4-x^2$ is shown below.

  

(c)

To determine

To describe: the intervals of increasing and decreasing.

Explanation of Solution

Given information:

Consider function $f(x)=2x^6+5x^4-x^2$

From part a,

Intervals of increasing $(-\infty ,-0.308)\cup (0,0.308)$

Intervals of decreasing $(-0.308,0)\cup (0.308,\infty )$

(d)

To determine

To find: whether the function is even, odd or neither.

Answer to Problem 14CT

The function is neither even.

Explanation of Solution

Given information:

Consider function $f(x)=2x^6+5x^4-x^2$

Calculation:

  $\begin{array}{l}f(x)=2x^6+5x^4-x^2\\ f(-x)=2{(}^{-}+5{(}^{-}-{(}^{-}\\ =2x^6+5x^4-x^2\\ =f(x)\end{array}$

Hence, the function is even.