Question

use the intermediate value theorem to show that the polynomial has a real zero between the given integers. ( f(x)=4 x^{3}-2 x - 5 ); between 1 and 3 select the correct choice below and, if necessary, fill in the answer box(es) within your choice. (simplify your answers.) a. because ( f(x) ) is a polynomial with ( f(1)=square<0 ) and ( f(3)=square>0 ), the function has a real zero between 1 and 3. b. because ( f(x) ) is a polynomial with ( f(1)=square<0 ) and ( f(3)=square<0 ), the function has a real zero between 1 and 3. c. because ( f(x) ) is a polynomial with ( f(1)=square>0 ) and ( f(3)=square>0 ), the function has a real zero between 1 and 3. d. because ( f(x) ) is a polynomial with ( f(1)=square>0 ) and ( f(3)=square<0 ), the function has a real zero between 1 and 3.

Explanation:

Step1: Calculate \(f(1)\)

Substitute \(x = 1\) into \(f(x)=4x^{3}-2x - 5\).

$$
LATEXBLOCK0
$$

Step2: Calculate \(f(3)\)

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

$$
LATEXBLOCK1
$$

Answer:

A. Because \(f(x)\) is a polynomial with \(f(1)=-3<0\) and \(f(3)=97>0\), the function has a real zero between \(1\) and \(3\).