Question

Medal anyone?
Can someone explain to me how you determine which intervals to use and which number should replace the "a" in f(a)? (files are attached)

Answer 1

End behavior of a function means what happens to y when \(x->\pm \infty\).

Answer 2

f(-3) means x should be replaced by -3 and the function evaluated.

Answer 3

I understand that the "f(-3) means x should be replaced by -3" but how was it decided that -3 should replace x or was it just randomly chosen?

Answer 4

To where is is positive or negative to make \[x ^{5}-5x ^{3}+4x = 0\]
and you will get 5 values for x
and then you will draw number lines to know where the function is negative and where it is positive with 5 numbers and substitute to know.

Answer 5

Any convenient number in the interval will be fine.

Answer 6

@Catseyeglint911 make the equation equal to zero and you will get these numbers u got me ?

Answer 7

So the intervals and the "a" in f(a) have nothing to do with each other?

Answer 8

The function is a 5th degree polynomial and it has 5 roots namely (from left to right):
-2, -1, 0, 1, 2. If you graph the function these are the points where it will cross the x -axis.

But we want to know where the function is positive or negative.

So we form 6 intervals: (-infinity, -2); (-2, -1); (-1, 0); (0, 1); (1, 2); (2, infinity).

We need a sample point in each interval that we can plug into the function and determine if the function is positive or negative in each interval.

Any convenient point is fine. They chose: -3, -1.5, -0.5, 0.5, 1.5 and 3

Answer 9

Each of those chosen point belongs to those six intervals.

Answer 10

the points are on x axes ,so you don't need to know if they are positive or negative

Answer 11

What defines a point as convenient?

Answer 12

Easy to evaluate f(x). A round number (integer), preferably a small number in magnitude.
In the first interval (-infinity, -2). any number to the LEFT of -2 is fine.
The next integer (round number) that is to the left of -2 is -3 and so they chose that for convenience.

In the next interval (-2, -1), we cannot choose an integer because there is none between -2 and -1. So they chose the midpoint -1.5

etc., etc.

Answer 13

Ahhh. Okay. Thank makes sense now.
And they chose those certain intervals based off of combining the different roots into coordinates?

Answer 14

convenient point is the point which you can easily substitute and get the answer like taking x = 0 or 1 some easy numbers, but any other number in the open interval will do the same job it will be positive or negative

Answer 15

exactly. At the roots we know the function is zero. But to the left or right of that root is the graph above the x-axis or below the x-axis? That is, is the function positive or negative? So they choose a convenient point in each interval and determine if f(x) is positive or negative.

Answer 16

And the intervals are determined by the roots.

Answer 17

Thank you so much for your explanations! @aum and @Catch.me

Answer 18

Would either one of you care to help me with a problem that deals with this information?

Answer 19

sure

Answer 20

You are welcome. Go ahead.

Answer 21

A polynomial function has a root of –6 with multiplicity 1, a root of –2 with multiplicity 3, a root of 0 with multiplicity 2, and a root of 4 with multiplicity 3. If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true?

~ The graph of the function is positive on (–6, –2).
~ The graph of the function is negative on (mc025-1.jpg, 0).
~ The graph of the function is positive on (–2, 4).
~ The graph of the function is negative on (4, mc025-2.jpg).

Answer 22

From the first paragraph, you can determine the polynomial is:
a(x+6)(x+2)^3x^2(x-4)^3 ---- (1)
where a is positive because leading coefficient is positive.

Now to determine IF The graph of the function is positive on (–6, –2) pick a convenient point in the interval (–6, –2). Let us pick -4.
Put -4 in (1) and determine if it is positive or negative. Note: We don't need the exact value of f(-4). Only if it is positive or negative.

Do the same with other answer choices.

Answer 23

(x+6)(x+2)^3x^2(x-4)^3
When x = -4:
(x+6) is positive
(x+2) is negative and so (x+2)^3 is negative
x^2 is always positive
(x-4)^3 is negative

+ x - x + x - = +

Answer 24

Okay I get what you've said so far.
So for (-infinity, 0) we could choose -3, and when x =-3:
(x+6) is positive
(x+2) is negative
(x+2)^3 is negative
x^2 is positive
(x-4)^3 is negative
So that answer would be incorrect. Right?

Answer 25

Yes, in (-infinity, 0), the function will be positive and so the second choice is false.

Also, since x^2 is always positive, leave it out and simply go with:
(x+6) * (x+2)^3 * (x-4)^3

Answer 26

For the third choice try x = 1.

Answer 27

x=1
(x+6) is positive
(x+2) is positive
(x+2)^3 is positive
x^2 is positive
(x-4)^3 is negative

Answer 28

yes, it is negative in that interval and so the third choice is false.

Answer 29

For the fourth choice x=5
(x+6) is positive
(x+2) is negative
(x+2)^3 is negative
x^2 is positive
(x-4)^3 is positive
and since this one is positive then it is incorrect?
Also, what makes the third one incorrect?

Answer 30

Yes, fourth choice is false.

To test in the interval (–2, 4) let us choose x = 1
(x+6)(x+2)^3x^2(x-4)^3
When x = 1:
(x+6) is positive
(x+2)^3 is positive
x^2 is positive
(x-4)^3 is negative

Therefore, the product is negative. And that makes the third choice false.

Answer 31

Thank you so very much for all of your help! :)

Answer 32

You are welcome.