Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the
given integers.
-$f ( x ) = 2 x ^ { 3 } - 8 x + 10$; between $- 3$ and $- 2$
A) $f ( - 3 ) = - 20$ and $f ( - 2 ) = 10$; yes
B) $f ( - 3 ) = - 20$ and $f ( - 2 ) = - 10$; no
C) $f ( - 3 ) = 20$ and $f ( - 2 ) = 10$; no
D) $f ( - 3 ) = 20$ and $f ( - 2 ) = - 10$; yes

Question

Quizplus · Accepted Answer

The Intermediate Value Theorem states that if a function $f$ is continuous on a closed interval $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there is at least one number $c$ in the interval $[a, b]$ such that $f(c) = N$. Calculating $f(-3) = 2(-3)^3 - 8(-3) + 10 = -54 + 24 + 10 = -20$ and $f(-2) = 2(-2)^3 - 8(-2) + 10 = -16 + 16 + 10 = 10$. Since $f(-3) = -20$ and $f(-2) = 10$, and these values have opposite signs, there must be at least one real zero between $-3$ and $-2$ according to the Intermediate Value Theorem.

Use the Intermediate Value Theorem to Determine Whether the Polynomial f(x)=2x3−8x+10f ( x ) = 2 x ^ { 3 } - 8 x + 10f(x)=2x3−8x+10

Use the Intermediate Value Theorem to Determine Whether the Polynomial $f ( x ) = 2 x ^ { 3 } - 8 x + 10$