We have the function:
[tex]F(x)=2x^3+2x^2-4[/tex]
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)
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow G(x)\rightarrow\infty \\ x\rightarrow-\infty\Rightarrow G(x)\rightarrow-\infty \end{gathered}[/tex]
This is TRUE and can be seen looking at the graph.
c)
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow F(x)\rightarrow\infty \\ x\rightarrow-\infty\Rightarrow F(x)\rightarrow-\infty \end{gathered}[/tex]
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.
[tex]\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}[/tex]
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.
