Which factor could be multiplied by the given function so that their product causes the graph of f(x) to decrease as x approaches negative infinity?

[tex]f(x)=(2x^2 + 5)(x - 7)[/tex]

Select all that apply.

• -3
• 0.2x
• 5x²
• [tex]-\frac{1}{2}(x + 8)[/tex]
• 4(2x - 7)

Answer:

5x²

-1/2( x+8)

Step-by-step explanation:

The dominant term is 2x^2 *x or 2x^3

We want to know what happens when x goes to negative ∞

2 ( -∞) ^3 → 2 * (-∞) → -∞

As long as we multiply by a positive number, we will still be approaching negative infinity ( or decreasing)

-3  is negative

.2x is negative when x approaches negative infinity

.5 x^2 = .5( -∞)^2 = 5 ( ∞) = ∞ which is positive

-1/2 x is the dominant term  = -1/2(- ∞) =   ∞ which is positive

4(2x) is the dominant term  = 8(- ∞) = - ∞ which is negative

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.  

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