Question

Based on the function F(x) = 2x8+2x2 - 4 and the graph of G(X) below, whichof the following statements is true?

Answer 1

We have the function:

`F(x)=2x^3+2x^2-4`

And a function G(x) in the graph.

We will analyze each statement:

a) G(x) has 3 real roots.

FALSE. We can see in the graph that the function has only one real root.

b)

`\begin{gathered} x\rightarrow\infty\Rightarrow G(x)\rightarrow\infty \\ x\rightarrow-\infty\Rightarrow G(x)\rightarrow-\infty \end{gathered}`

This is TRUE and can be seen looking at the graph.

c)

`\begin{gathered} x\rightarrow\infty\Rightarrow F(x)\rightarrow\infty \\ x\rightarrow-\infty\Rightarrow F(x)\rightarrow-\infty \end{gathered}`

If we increase x, the predominant term for F(x) is 2x^3, that is positive and will tend to infinity as x tends to infinity, and will tend to minus infinity as x tends to minus infinity.

Then, this is TRUE.

d) F(x) has 3 real roots.

We have to factorize in order to see the roots of the function.

`\begin{gathered} 2x^3+2x^2-4=0 \\ 2(x^3+x^2-2)=0 \\ 2(x-1)(x^2+x^2+2)=0 \end{gathered}`

The only real root for this function is x=1.

The other two are complex roots.

This statement is FALSE.

The statements that are true are B and C.