Answer:
3rd and 4th Choices.
[tex]\displaystyle 5x^2\text{ and } -\frac{1}{2}(x+8)[/tex]
Step-by-step explanation:
We are given the function:
[tex]f(x)=(2x^2+5)(x-7)[/tex]
And we want to determine the factor(s) that we can multiply to the above function that will cause the graph of f to decrease as x approaches negative infinity.
We can see that the dominant term will be:
[tex]2x^2(x)=2x^3[/tex]
Thus:
[tex]2(-\infty)^3\Rightarrow -\infty[/tex]
It is already decreasing (shifting towards negative infinity) as x approaches negative infinity.
In order to preserve this, then, the factors must be positive as x approaches negative infinity.
The first factor is -3.
-3 is always negative, so it will make f increase.
The second factor is [tex]0.2x[/tex].
[tex]0.2(-\infty)\Rightarrow -\infty[/tex]
This will also make f increase.
The third factor is:
[tex]5(\infty)^2\Rightarrow \infty[/tex]
This is positive, so it will allow f to remain decreasing as x approaches negative infinity.
The fourth factor is:
[tex]\displaystyle -\frac{1}{2}(-\infty+8)\Rightarrow -\frac{1}{2}(-\infty)\Rightarrow \infty[/tex]
This is also positive, so it will also allow f to remain decreasing.
Lastly, the fifth factor is:
[tex]4(2(-\infty)-7)\Rightarrow 4(-\infty)\Rightarrow -\infty[/tex]
This is negative, so it will make f increasing.
Therefore, our answers are the third and fourth choices.
