Thread: Solving a polynomial problem without guess & check!

Hey everyone,

I have the following problem and I know I could solve with guess & check, but how would I solve it to be able to prove my answer?


Problem:

Find the range of values of 'c' such that the function f(x) = 2x3 - x2 - 6x + c has a zero between x = 0 and x = 1





Thanks everyone!

Originally Posted by qcom

Hey everyone,

I have the following problem and I know I could solve with guess & check, but how would I solve it to be able to prove my answer?

Problem:

Find the range of values of 'c' such that the function f(x) = 2x^3 - x^2 - 6x + c has a zero between x = 0 and x = 1

in future, use the caret symbol ^ to indicate exponents, or learn Latex.









for a zero to exist between and , there has to be a sign change of the function values

if , then




looking at the other possibility ...

if , then

looks like that possibility can't happen ... do you see why?

Sorry for the messy exponents, what is Latex by the way?

Now, I understood how you logically proceeded with the problem, now I think it's not possible because the question calls for the zeros to be between 0 and 1 and that would require a sign change, yet a sign change is impossible between those two integers?

Originally Posted by qcom

Sorry for the messy exponents, what is Latex by the way?

Now, I understood how you logically proceeded with the problem, now I think it's not possible because the question calls for the zeros to be between 0 and 1 and that would require a sign change, yet a sign change is impossible between those two integers?

I'm afraid you don't understand ... I checked two possible cases.

Case #1 : f(0) positive and f(1) negative. this case worked as shown below ...







for a zero to exist between and , there has to be a sign change of the function values ...

if , then




Case #2 : f(0) negative and f(1) positive does not work.


I suggest you graph f(x) using values of c between 0 and 5 and confirm for yourself that a zero for f(x) occurs between x = 0 and x = 1.

OK, I see where I was confused, thanks for your help and patience!