Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-$$f ( x ) = 5 x ^ { 3 } - 7 x ^ { 2 } + 4 x - 7 ; [ 1,2 ]$$
A) $f ( 1 ) = - 5$ and $f ( 2 ) = 13$; yes
B) $f ( 1 ) = 5$ and $f ( 2 ) = - 13$; yes
C) $f ( 1 ) = 5$ and $f ( 2 ) = 13$; no
D) $f ( 1 ) = - 5$ and $f ( 2 ) = - 13$; no

Question

Quizplus · Accepted Answer

The Intermediate Value Theorem states that if $f$ is continuous on a closed interval $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the interval $[a, b]$ such that $f(c) = N$. Calculating $f(1) = 5(1)^3 - 7(1)^2 + 4(1) - 7 = -5$ and $f(2) = 5(2)^3 - 7(2)^2 + 4(2) - 7 = 13$, we see that the function values at the endpoints of the interval $[1, 2]$ are of opposite signs, indicating a zero must exist between them.

Use the Intermediate Value Theorem to Determine Whether the Polynomial f(x)=5x3−7x2+4x−7;[1,2]f ( x ) = 5 x ^ { 3 } - 7 x ^ { 2 } + 4 x - 7 ; [ 1,2 ]f(x)=5x3−7x2+4x−7;[1,2]

Use the Intermediate Value Theorem to Determine Whether the Polynomial $f ( x ) = 5 x ^ { 3 } - 7 x ^ { 2 } + 4 x - 7 ; [ 1,2 ]$