anonymous
  • anonymous
f(x)=(x^3)-(x^2)+x-2 [0,3] f(c)=4 I already found what f(0) and f(3) are
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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dape
  • dape
What is the question?
jdoe0001
  • jdoe0001
do you know the rational root test? because I'd think you'd need it
anonymous
  • anonymous
"Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of 'c' guaranteed by the theorem".

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anonymous
  • anonymous
& I don't need to use the rational root test..
dape
  • dape
Do you know what the intermediate value theorem states?
anonymous
  • anonymous
If f is continuous on the closed interval between [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
dape
  • dape
Yup, so first of all, is f(c)=4 between f(a) and f(b)?
anonymous
  • anonymous
Yeah
dape
  • dape
So then the intermediate value theorem guarantees that f(c)=4 has a solution for some c.
dape
  • dape
So now, if you put in c for x in the function and put it equal to 4 (f(c)=4) and solve for c, you will have your answer.
anonymous
  • anonymous
Right now i have 4=x^3-x^2+x-2, should I subtract the 4 over before substituting in c?
dape
  • dape
The equation might have more than one solution, but we are only interested of the solution(s) in the interval [0,3].
dape
  • dape
You should substitute c before, if you want to be thorough, although it doesn't really matter as long as you keep track of your variable names.
dape
  • dape
On an exam you would want to use c in the equation, so 4=c^3-c^2+c-2
anonymous
  • anonymous
okay now I have 4=c^3-c^2+c-2
dape
  • dape
Do you know how to solve it?
anonymous
  • anonymous
yeah lol
dape
  • dape
Then solve it and you will have your answer :)
anonymous
  • anonymous
okay now I have 4=c^3-c^2+c-2
anonymous
  • anonymous
@dape i actually don't know what to do from here lmao
dape
  • dape
Start by subtracting 4 from both sides.
anonymous
  • anonymous
0= c^3 - c^2 + c - 6
dape
  • dape
Yes, now since the coefficient in front of the highest order term (\(c^3\)) is 1, any rational solutions must be a divisor of the constant term \(-6\). So you can guess the solution by putting in different factors of \(-6\), one is bound to be a solution. Also the intermediate value theorem guarantees that the solution lies in [0,3], so check which factor of \(-6\) between 0 and 3 is a solution.
dape
  • dape
Do you understand this?
dape
  • dape
In other words, when you have an equation like that with whole number coefficients and a 1 in front of the highest order term you can always guess solutions from the factors of the constant term.
anonymous
  • anonymous
I don't understand o_o
dape
  • dape
When you see a polynomial equation (like yours) with only whole numbers as coefficients it's a good idea to guess what the solution is. The right answer must be a factor of the constant term, in your case the constant term is \(-6\), the factors of \(-6\) is \[\pm1,\pm2,\pm3,\pm6\] You could put in each of these 8 numbers to find which one solves the equation. But you also know from the intermediate value theorem that a solution must exist between 0 and 3 (the interval in the problem), so you only have to consider \[+1,+2,+3\] Now one of these three must be a solution, try each and you will find the answer.
anonymous
  • anonymous
where did you get those numbers from?
dape
  • dape
They are factors of \(-6\), you can multiply them together to get \(-6\).
anonymous
  • anonymous
OH Okay I see where the numbers came from, how did you know to use the factors of 6 though?
anonymous
  • anonymous
@dape
dape
  • dape
Read carefully what I wrote above, essentially it's a sophisticated form of guess.
anonymous
  • anonymous
okay I got f(1) = -5 f(2) = 0 f(3) = 15 but my book says the answer is f(2) = 4 ._.
anonymous
  • anonymous
@dape
dape
  • dape
If you put 2 into \(c^3 - c^2 + c - 6\) you get 0, so it solves our equation. This means that \(f(2)=4\) as you can check. Try putting 2 into the function and you will see that the book is correct.
anonymous
  • anonymous
OH BECAUSE I WAS SOLVING FOR C
anonymous
  • anonymous
i'm so brain dead

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