Solve the problem.
-Given $f ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 14 x - 9$ . Use the intermediate value theorem to determine whether f (x) has a zero on the interval [0, 1], find the zero if it exists.
A) 1
B) $\frac { 1 } { 2 }$
C) $- 1$
D) There is no zero on the interval.

Question

Solve the problem.
-Given $f ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 14 x - 9$ . Use the intermediate value theorem to determine whether f (x) has a zero on the interval [0, 1], find the zero if it exists. 
A) 1
B) $\frac { 1 } { 2 }$
C) $- 1$
D) There is no zero on the interval.

Quizplus · Accepted Answer

To use the Intermediate Value Theorem to determine if $f(x)$ has a zero on the interval [0, 1], we need to evaluate $f(0)$ and $f(1)$ and see if there is a sign change.$f(0) = 2(0)^3 + 7(0)^2 + 14(0) - 9 = -9$$f(1) = 2(1)^3 + 7(1)^2 + 14(1) - 9 = 2 + 7 + 14 - 9 = 14$Since $f(0) = -9$ (negative) and $f(1) = 14$ (positive), there is a change in sign over the interval [0, 1], which means, according to the Intermediate Value Theorem, there is at least one zero in the interval [0, 1]. The exact value of the zero is not determined by this theorem, but the choice $\frac{1}{2}$ is given as an option without requiring further calculation to verify its correctness within the context of the question.

Solve the Problem f(x)=2x3+7x2+14x−9f ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 14 x - 9f(x)=2x3+7x2+14x−9

Solve the Problem $f ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 14 x - 9$