Question

using the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.
f(x)=x^{4}-5x^{2}-5; a = 3, b = 5

select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
oa. by the intermediate value theorem, the function has at least one real zero between a and b because f(a) = and f(b) = (simplify your answers.)
ob. by the intermediate value theorem, the function does not have at least one real zero between a and b because f(a) = and f(b) = (simplify your answers.)
oc. it is impossible to use the intermediate value theorem in this case.

Explanation:

Step1: Calculate \( f(a) \)

Substitute \( a = 3 \) into \( f(x)=x^{4}-5x^{2}-5 \).

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Step2: Calculate \( f(b) \)

Substitute \( b = 5 \) into \( f(x)=x^{4}-5x^{2}-5 \).

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Since \( f(x)=x^{4}-5x^{2}-5 \) is a polynomial function, it is continuous everywhere. And \( f(3)=31>0 \), \( f(5)=495>0 \). The Intermediate - Value Theorem states that if \( y = f(x) \) is continuous on a closed interval \([a,b]\) and \( k\) is a number between \( f(a)\) and \( f(b)\), then there exists at least one number \( c\in(a,b)\) such that \( f(c)=k\). But when \( f(a)\) and \( f(b)\) have the same sign, we cannot guarantee that there is a zero (i.e., \( k = 0\)) between \( a\) and \( b\) using the Intermediate - Value Theorem.

Answer:

C. It is impossible to use the intermediate value theorem in this case.