Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-$$f ( x ) = 7 x ^ { 3 } - 2 x + 2 ; [ - 1,0 ]$$
A) $f ( - 1 ) = 3$ and $f ( 0 ) = - 2$; yes
B) $f ( - 1 ) = 3$ and $f ( 0 ) = 2$; no
C) $f ( - 1 ) = - 3$ and $f ( 0 ) = 2$; yes
D) $f ( - 1 ) = - 3$ and $f ( 0 ) = - 2$; no

Question

Quizplus · Accepted Answer

First, calculate $f(-1) = 7(-1)^3 - 2(-1) + 2 = -7 + 2 + 2 = -3$ and $f(0) = 7(0)^3 - 2(0) + 2 = 2$. Since $f(-1) = -3$ and $f(0) = 2$, and these values have opposite signs, the Intermediate Value Theorem guarantees a zero in the interval $[-1, 0]$.

Use the Intermediate Value Theorem to Determine Whether the Polynomial f(x)=7x3−2x+2;[−1,0]f ( x ) = 7 x ^ { 3 } - 2 x + 2 ; [ - 1,0 ]f(x)=7x3−2x+2;[−1,0]

Use the Intermediate Value Theorem to Determine Whether the Polynomial $f ( x ) = 7 x ^ { 3 } - 2 x + 2 ; [ - 1,0 ]$