Question

Find all values of x such that f(x) >0, all x such that f(x) <0?
#1 f(x)=-1/16x^4+1
#2 f(x)=1/4x^3-2 [I kind of already solved this one sorry]
Answer in text book = f(x)>0 if IxI < 2, f(x)<0 if IxI > 2

Can someone please explain to me how they got that answer? I really have no clue where the absolute value came from.

Answer 1

i tried solving for the inverse of the function then algebraically manipulated it so that its comes out to this : 4x^3 (y-2) =1 . I plugged in y=0 and solved for x/ however the answer comes out to x =1/2 .. close but not the answer you have

Answer 2

nevermind, you changed the problem

Answer 3

First you have find the intervals on which f(x) will be greater than 0.
Find them by factoring the function first.

Give me a sec ,I'll write it on the paper & I'll show it to you.

Answer 4

Mrhoola i'm very sorry I changed it. I noticed that you were on this question for a good amount of time and for that I really appreciate your efforts.

Answer 5

tyvm Jadzia

Answer 6

\(\sf f(x) > 0 \\-1/16 x^4+1 > 0\\-1/16(x^4-16)>0\\-1/16(x^2+4)(x^2-4)>0\\-1/16(x^2+4)(x+2)(x-2)>0\)

So now from the factors, you can test the sign of the function using the intervals:

\(\sf x < -2~~ |\ -2 <x < 2~~|\ 2 <x \)


Get some test values from these intervals to determine which one of them will give f(x) > 0 . Do you know how to do it?

Answer 7

I'm a bit confused on the 2nd step that you did. How did (x^4-16) come to be? And to answer your question, no I do not i'm very sorry. Tyvm for the help so far though :)

Answer 8

I factored out -1/16 from the function, so that we can easily find the zeros.
That's alright (: We can do it step by step. Ask me if you still have any other questions about what I did above so that we can clear that up and move to the next step ^_^

Answer 9

Oh that makes so much sense now. I get everything you did so far above. Now it's on to the test values I assume?

Answer 10

Opps one more question, how did you determine the intervals? I'm assuming that x<-2 came from (x+2) and that 2<x came from (x-2)?

Answer 11

Great (: Yes that is right.

Now let's take any test values from the intervals.
I'll try to do the first interval:
x < -2 , we can take x = -3

now the factors of the function are:
\(\sf -1/16\) this will make f(x) NEGATIVE
\(\sf (x^2 + 4)\) , substituting x=-3, you'll get a POSITIVE number
\(\sf (x+2)\), substituting x=-3, you'll get a NEGATIVE number
\(\sf (x-2)\) substituting x=-3, you'll get a NEGATIVE number

Now a negative x positive x negative x negative will give you a negative value, so f(x) < 0 when x <-2. We have to do the same method with the two other intervals.

Answer 12

Okay, so you just 2 other numbers than?

Answer 13

then*

Answer 14

plug* omg i'm so sorry

Answer 15

I might need to refresh my browser, but here is what I did...

Answer 16

Okay that make's sense. Was there any reason that you chose 3,0 and -3 as the test values? And what about the book's answer? = f(x)>0 if IxI < 2, f(x)<0 if IxI > 2
Are both of the answers the same, but written differently?

Answer 17

the test values are any values within the interval, say for example, x < -2 .. you can pick any test values that are less than -2 . We really don't care about the values that it will give, we are only focusing on the resulting "sign"

Answer 18

the book only uses absolute value to represent it in a simple way, but they are the same thing. I just wrote that way so that you can understand it better

Answer 19

Okay makes sense now.

Answer 20

I think i'm just getting confused about the absolute value signs in the book's answer now. Is there more to be done or are the answers the same, but written differently?

Answer 21

okay let's try to evaluate what the book say: f(x)>0 if IxI < 2

|x| < 2 is the same as:

-x < 2 or x > -2

AND

x < 2

hence, if you combine those two, you'll get the interval -2 < x < 2

Answer 22

I'm really sorry, i'm not quite following. Where did the -2 come from in -x < 2 or x> -2?

Answer 23

I get because of of the absolute value, there has to be -x and x

Answer 24

yes, you are right.. |x| will be -x AND x

|x| < 2 --> this is the interval from the book

Since |x| will be -x AND x, you will end up with two "intervals"
so you will have:

-x < 2 AND x < 2

which will be the same as -2 < x < 2
we will get -2 from this interval : -x < 2
multiply both sides by "-1 " in order make "x" positive, but multiplying/dividing an inequality by any negative number will switch the direction of the inequality sign in the opposite direction.

Answer 25

Okay thank you so much for clearing that up and sticking this long with my questions :D. You have officially put and end to my almost 3 hour search for the answer to this problem and for that I am really grateful <3

Answer 26

No problem. Pleasure to help :) Btw, Welcome to Openstudy !