use the intermediate value theorem to determine whether the polynomial function has a real zero between the…

use the intermediate value theorem to determine whether the polynomial function has a real zero between the given integers. f(x)= - 8x^4 + 5x^2 + 4; between - 2 and - 1\n\na. f(-2)= - 104 and f(-1)=1; yes\nb. f(-2)=104 and f(-1)= - 1; yes\nc. f(-2)=104 and f(-1)=2; no\nd. f(-2)= - 104 and f(-1)= - 1; no

use the intermediate value theorem to determine whether the polynomial function has a real zero between the given integers. f(x)= - 8x^4 + 5x^2 + 4; between - 2 and - 1\n\na. f(-2)= - 104 and f(-1)=1; yes\nb. f(-2)=104 and f(-1)= - 1; yes\nc. f(-2)=104 and f(-1)=2; no\nd. f(-2)= - 104 and f(-1)= - 1; no

Answer

Explanation:

Step1: Calculate f(-2)

Substitute x = - 2 into f(x)=-8x^{4}+5x^{2}+4. [ \begin{align*} f(-2)&=-8\times(-2)^{4}+5\times(-2)^{2}+4\ &=-8\times16 + 5\times4+4\ &=-128+20 + 4\ &=-104 \end{align*} ]

Step2: Calculate f(-1)

Substitute x=-1 into f(x)=-8x^{4}+5x^{2}+4. [ \begin{align*} f(-1)&=-8\times(-1)^{4}+5\times(-1)^{2}+4\ &=-8\times1+5\times1 + 4\ &=-8 + 5+4\ &=1 \end{align*} ]

Step3: Apply Intermediate - Value Theorem

Since f(x) is a polynomial function, it is continuous everywhere. f(-2)=-104<0 and f(-1)=1>0. By the Intermediate - Value Theorem, if a function 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 the interval (a,b) such that f(c)=k. In the case of a zero, k = 0. Since 0 is between f(-2) and f(-1), there is a real zero between - 2 and - 1.

Answer:

A. f(-2)=-104 and f(-1)=1; yes