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Existence of a polynomial.
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I have little progress while working on this STEP question.
I have to determine whether:
There exists a polynomial P such that |P(x) - cos x| <= 10^-6.
I have tried to write the LHS as a piecewise defined function i.e.
Let f(x) = |P(x) - cos|
Case 1:
P(x)<cos x --> cos x - P(x) <= 10^-6
Case 2:
P(x)>=cos x --> P(x) -cos x<=10^-6.
Other than that I am not sure how to continue on this, as I am not given any other information.
Can someone please provide me with hints on how to continue/ what approach I should take for this problem?
I have to determine whether:
There exists a polynomial P such that |P(x) - cos x| <= 10^-6.
I have tried to write the LHS as a piecewise defined function i.e.
Let f(x) = |P(x) - cos|
Case 1:
P(x)<cos x --> cos x - P(x) <= 10^-6
Case 2:
P(x)>=cos x --> P(x) -cos x<=10^-6.
Other than that I am not sure how to continue on this, as I am not given any other information.
Can someone please provide me with hints on how to continue/ what approach I should take for this problem?
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#2
Perhaps I'm misunderstanding or there's more to the question (this doesn't seem difficult enough for STEP as is), but surely for any polynomial |P(x)-cos(x)| tends to infinity as x tends to infinity (P unbounded, cos bounded) so there isn't such a polynomial.
Unless you're including power series in which case you have your answer to an even higher accuracy.
Unless you're including power series in which case you have your answer to an even higher accuracy.
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#3
(Original post by ThomH97)
Perhaps I'm misunderstanding or there's more to the question (this doesn't seem difficult enough for STEP as is), but surely for any polynomial |P(x)-cos(x)| tends to infinity as x tends to infinity (P unbounded, cos bounded) so there isn't such a polynomial.
Unless you're including power series in which case you have your answer to an even higher accuracy.
Perhaps I'm misunderstanding or there's more to the question (this doesn't seem difficult enough for STEP as is), but surely for any polynomial |P(x)-cos(x)| tends to infinity as x tends to infinity (P unbounded, cos bounded) so there isn't such a polynomial.
Unless you're including power series in which case you have your answer to an even higher accuracy.
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