Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$ 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

Quizplus · Accepted Answer

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 zeros beyond a certain point for polynomials. To determine whether a polynomial has no real zero greater than a specific value, one would typically use other methods, such as analyzing the behavior of the polynomial or applying the Descartes' Rule of Signs. The statement in question misrepresents the application of the boundedness theorem to the problem of finding zeros of a polynomial outside a given interval.

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 $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$