Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial f(x) $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$ 4x has no real zero greater than 2.
A) Yes, the boundedness theorem shows that the polynomial has no real zero greater than 2.
B) No, the boundedness theorem does not show that the polynomial has no real zero greater than 2.

Question

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The boundedness theorem, also known as the boundedness principle or extreme value theorem, states that a continuous function on a closed interval [a, b] attains its maximum and minimum values, at least once each, within that interval. However, it does not provide information about the existence or non-existence of real zeros beyond a certain point for polynomials. The question of whether a polynomial has no real zero greater than a certain value cannot be directly answered by the boundedness theorem. The behavior of the polynomial $f(x) = x^5 - 5x^3 + 4x$ for values greater than 2 involves analyzing its increasing or decreasing nature and its limits at infinity, which are not covered by the boundedness theorem.

Use the Boundedness Theorem to Determine Whether the Polynomial Function f(x)=x5−5x3+4xf ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 xf(x)=x5−5x3+4x

Use the Boundedness Theorem to Determine Whether the Polynomial Function $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$