Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given
integers
-$f ( x ) = 7 x ^ { 3 } - 4 x ^ { 2 } + 9 x + 3$; between $- 1$ and 0
A) $f ( - 1 ) = 17$ and $f ( 0 ) = 3$; no
B) $f ( - 1 ) = - 17$ and $f ( 0 ) = - 3$; no
C) $f ( - 1 ) = - 17$ and $f ( 0 ) = 3$; yes
D) $f ( - 1 ) = 17$ and $f ( 0 ) = - 3$; 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 exists at least one number $c$ in the interval $[a, b]$ such that $f(c) = N$. For $f(x) = 7x^3 - 4x^2 + 9x + 3$, calculating $f(-1)$ gives $-17$ and $f(0)$ gives $3$. Since the signs of $f(-1)$ and $f(0)$ are different, there must be at least one real zero between $-1$ and $0$, according to the Intermediate Value Theorem.

Use the Intermediate Value Theorem to Determine Whether the Polynomial f(x)=7x3−4x2+9x+3f ( x ) = 7 x ^ { 3 } - 4 x ^ { 2 } + 9 x + 3f(x)=7x3−4x2+9x+3

Use the Intermediate Value Theorem to Determine Whether the Polynomial $f ( x ) = 7 x ^ { 3 } - 4 x ^ { 2 } + 9 x + 3$