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Let
g be a continuous function on the closed interval
[-1,3], where
g(-1)=-2 and
g(3)=-5.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
g(c)=-3 for at least one
c between -1 and 3
(B)
g(c)=0 for at least one
c between -5 and -2
(C)
g(c)=-3 for at least one
c between -5 and -2
(D)
g(c)=0 for at least one
c between -1 and 3

Let $g$ be a continuous function on the closed interval $[-1,3]$ , where $g(-1)=-2$ and $g(3)=-5$ . $\newline$ 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 $-1$ and $3$ $\newline$ (B) $g(c)=0$ for at least one $c$ between $-5$ and $-2$ $\newline$ (C) $g(c)=-3$ for at least one $c$ between $-5$ and $-2$ $\newline$ (D) $g(c)=0$ for at least one $c$ between $-1$ and $3$

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Intermediate Value Theorem

Full solution

Q. Let

g

be a continuous function on the closed interval

[-1,3]

, where

g(-1)=-2

and

g(3)=-5

.

\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

-1

and

3

\newline

(B)

g(c)=0

for at least one

c

between

-5

and

-2

\newline

(C)

g(c)=-3

for at least one

c

between

-5

and

-2

\newline

(D)

g(c)=0

for at least one

c

between

-1

and

3

Intermediate Value Theorem: 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 $[-1, 3]$ , $g(-1) = -2$ , and $N$ $0$ .
Option Analysis (A): We need to determine which of the given options is guaranteed by the Intermediate Value Theorem. Let's analyze each option: $\newline$ (A) $g(c) = -3$ for at least one $c$ between $-1$ and $3$ . $\newline$ Since $-3$ is between $g(-1) = -2$ and $g(3) = -5$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $[-1, 3]$ such that $g(c) = -3$ .
Option Analysis (B): (B) $g(c) = 0$ for at least one $c$ between $-5$ and $-2$ . This option is not relevant to the Intermediate Value Theorem because it refers to values between $g(-1)$ and $g(3)$ , not to values of $c$ in the interval $[-1, 3]$ .
Option Analysis (C): $(C) g(c) = -3$ for at least one $c$ between $-5$ and $-2$ . This option is also not relevant to the Intermediate Value Theorem because it refers to values between $g(-1)$ and $g(3)$ , not to values of $c$ in the interval $[-1, 3]$ .
Option Analysis (D): (D) $g(c) = 0$ for at least one $c$ between $-1$ and $3$ . Since $0$ is not between $g(-1) = -2$ and $g(3) = -5$ , the Intermediate Value Theorem does not guarantee that there is a $c$ in the interval $[-1, 3]$ such that $g(c) = 0$ .

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\newline

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

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Question

g

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g(c)=3

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Question

g

be a continuous function on the closed interval

[-1,3]

g(-1)=-2

g(3)=-5

\newline

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

\newline

1

\newline

g(c)=-3

for at least one

c

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-2

\newline

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Question

f

be a continuous function on the closed interval

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\newline

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

\newline

1

\newline

f(c)=0

for at least one

c

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1

\newline

f(c)=0

for at least one

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\newline

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\newline

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\newline

Can we use the intermediate value theorem to say the equation

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1

\newline

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h

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\newline

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1

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