Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-$$f ( x ) = - 6 x ^ { 4 } - 3 x ^ { 2 } + 6 ; [ - 1,0 ]$$
A) $f ( - 1 ) = 3$ and $f ( 0 ) = 7$; no
B) $f ( - 1 ) = - 3$ and $f ( 0 ) = - 6$; no
C) $f ( - 1 ) = 3$ and $f ( 0 ) = - 6$; yes
D) $f ( - 1 ) = - 3$ and $f ( 0 ) = 6$; yes

Question

Quizplus · Accepted Answer

To determine if a polynomial function has a zero in the given interval using the Intermediate Value Theorem, we need to evaluate the function at the endpoints of the interval and see if there is a sign change. Evaluating $f(x) = -6x^4 - 3x^2 + 6$ at $x = -1$ and $x = 0$, we get $f(-1) = -6(-1)^4 - 3(-1)^2 + 6 = -6 - 3 + 6 = -3$ and $f(0) = -6(0)^4 - 3(0)^2 + 6 = 6$. Since there is a sign change from negative to positive, the function must have at least one zero in the interval $[-1, 0]$.

Use the Intermediate Value Theorem to Determine Whether the Polynomial f(x)=−6x4−3x2+6;[−1,0]f ( x ) = - 6 x ^ { 4 } - 3 x ^ { 2 } + 6 ; [ - 1,0 ]f(x)=−6x4−3x2+6;[−1,0]

Use the Intermediate Value Theorem to Determine Whether the Polynomial $f ( x ) = - 6 x ^ { 4 } - 3 x ^ { 2 } + 6 ; [ - 1,0 ]$