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Let
g be a continuous function on the closed interval
[-1,4], where
g(-1)=-4 and
g(4)=1.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
g(c)=3 for at least one
c between -4 and 1
(B)
g(c)=3 for at least one
c between -1 and 4
(C)
g(c)=-3 for at least one
c between -1 and 4
(D)
g(c)=-3 for at least one
c between -4 and 1

Let $g$ be a continuous function on the closed interval $[-1,4]$ , where $g(-1)=-4$ and $g(4)=1$ . $\newline$ Which of the following is guaranteed by the Intermediate Value Theorem? $\newline$ Choose $1$ answer: $\newline$ (A) $g(c)=3$ for at least one $c$ between $-4$ and $1$ $\newline$ (B) $g(c)=3$ for at least one $c$ between $-1$ and $4$ $\newline$ (C) $g(c)=-3$ for at least one $c$ between $-1$ and $4$ $\newline$ (D) $g(c)=-3$ for at least one $c$ between $-4$ and $1$

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Math Problems

Calculus

Intermediate Value Theorem

Full solution

Q. Let

g

be a continuous function on the closed interval

[-1,4]

, where

g(-1)=-4

and

g(4)=1

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

Choose

1

answer:

\newline

(A)

g(c)=3

for at least one

c

between

-4

and

1

\newline

(B)

g(c)=3

for at least one

c

between

-1

and

4

\newline

(C)

g(c)=-3

for at least one

c

between

-1

and

4

\newline

(D)

g(c)=-3

for at least one

c

between

-4

and

1

Introduction: The Intermediate Value Theorem states that if a function 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 $c$ in the interval $[a, b]$ such that $f(c) = N$ . We are given that $g$ is continuous on the interval $[-1, 4]$ , $g(-1) = -4$ , and $N$ $0$ .
IVT Explanation: We need to determine which of the given statements is guaranteed by the Intermediate Value Theorem. Let's analyze each option: $\newline$ (A) $g(c) = 3$ for at least one $c$ between $-4$ and $1$ . This option is not relevant because the interval for $c$ should be between the $x$ -values $-1$ and $4$ , not the $y$ -values $-4$ and $1$ .
Option Analysis: (B) $g(c) = 3$ for at least one $c$ between $-1$ and $4$ . Since $g(-1) = -4$ and $g(4) = 1$ , and $3$ is between $-4$ and $1$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $c$ $0$ such that $g(c) = 3$ .
Option A: $(C) g(c) = -3$ for at least one $c$ between $-1$ and $4$ . Since $g(-1) = -4$ and $g(4) = 1$ , and $-3$ is between $-4$ and $1$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $c$ $0$ such that $c$ $1$ .
Option B: $(D) g(c) = -3$ for at least one $c$ between $-4$ and $1$ . This option, like option $(A)$ , is not relevant because the interval for $c$ should be between the $x$ -values $-1$ and $4$ , not the $y$ -values $-4$ and $1$ .
Option C: Comparing options (B) and (C), we see that both are correct according to the Intermediate Value Theorem. However, the question asks us to choose one answer. Since the options in the problem statement might be exclusive, there might be an error in the problem statement or the options provided. In a typical scenario, only one of these options would be correct based on the values given for $g(-1)$ and $g(4)$ . If we assume that the problem statement is correct and only one answer is to be chosen, we would need additional information to determine which of the two is the correct answer.