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Let
g be a continuous function on the closed interval
[-3,3], where
g(-3)=0 and
g(3)=6.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
g(c)=-2 for at least one
c between 0 and 6
(B)
g(c)=5 for at least one
c between 0 and 6
(C)
g(c)=5 for at least one
c between -3 and 3
(D)
g(c)=-2 for at least one
c between -3 and 3

Let $g$ be a continuous function on the closed interval $[-3,3]$ , where $g(-3)=0$ and $g(3)=6$ . $\newline$ Which of the following is guaranteed by the Intermediate Value Theorem? $\newline$ Choose $1$ answer: $\newline$ (A) $g(c)=-2$ for at least one $c$ between $0$ and $6$ $\newline$ (B) $g(c)=5$ for at least one $c$ between $0$ and $6$ $\newline$ (C) $g(c)=5$ for at least one $c$ between $-3$ and $3$ $\newline$ (D) $g(c)=-2$ for at least one $c$ between $-3$ and $3$

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Intermediate Value Theorem

Full solution

Q. Let

g

be a continuous function on the closed interval

[-3,3]

, where

g(-3)=0

and

g(3)=6

.

\newline

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

\newline

Choose

1

answer:

\newline

(A)

g(c)=-2

for at least one

c

between

0

and

6

\newline

(B)

g(c)=5

for at least one

c

between

0

and

6

\newline

(C)

g(c)=5

for at least one

c

between

-3

and

3

\newline

(D)

g(c)=-2

for at least one

c

between

-3

and

3

Apply 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 need to apply this theorem to the function $g$ , which is continuous on the interval $[-3, 3]$ , with $g(-3) = 0$ and $N$ $0$ .
Examine Option (A): We examine option (A): $g(c) = -2$ for at least one $c$ between $0$ and $6$ . This option is not relevant to the Intermediate Value Theorem because the interval $[0, 6]$ is not the domain of the function $g$ ; the domain given is $[-3, 3]$ . Additionally, $-2$ is not between the values of $g(-3)$ and $g(3)$ .
Examine Option (B): We examine option (B): $g(c) = 5$ for at least one $c$ between $0$ and $6$ . Similar to option (A), the interval $[0, 6]$ is not the domain of the function $g$ , and thus this option is not relevant to the Intermediate Value Theorem.
Examine Option (C): We examine option (C): $g(c) = 5$ for at least one $c$ between $-3$ and $3$ . Since $5$ is a number between $g(-3) = 0$ and $g(3) = 6$ , and the function $g$ is continuous on the interval $[-3, 3]$ , the Intermediate Value Theorem guarantees that there exists at least one $c$ in the interval $[-3, 3]$ such that $g(c) = 5$ .
Examine Option (D): We examine option (D): $g(c) = -2$ for at least one $c$ between $-3$ and $3$ . Since $-2$ is not between $g(-3) = 0$ and $g(3) = 6$ , the Intermediate Value Theorem does not guarantee that there exists a $c$ in the interval $[-3, 3]$ such that $g(c) = -2$ .

More problems from Intermediate Value Theorem

Question

f

be a continuous function on the closed interval

[1,5]

f(1)=1

f(5)=-3

\newline

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

\newline

1

\newline

f(c)=2

for at least one

c

-3

1

\newline

f(c)=-2

for at least one

c

-3

1

\newline

f(c)=2

for at least one

c

1

5

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f(c)=-2

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Question

h

be a continuous function on the closed interval

[-3,4]

h(-3)=-1

h(4)=2

\newline

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

\newline

1

\newline

h(c)=1

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-1

2

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h(c)=1

for at least one

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Posted 3 months ago

Question

g

be a continuous function on the closed interval

[-3,3]

g(-3)=0

g(3)=6

\newline

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

\newline

1

\newline

g(c)=-2

for at least one

c

-3

3

\newline

g(c)=-2

for at least one

c

0

6

\newline

g(c)=5

for at least one

c

-3

3

\newline

g(c)=5

for at least one

c

0

6

Posted 3 months ago

Question

g

be a continuous function on the closed interval

[-1,4]

g(-1)=-4

g(4)=1

\newline

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

\newline

1

\newline

g(c)=3

for at least one

c

-1

4

\newline

g(c)=-3

for at least one

c

-4

1

\newline

g(c)=3

for at least one

c

-4

1

\newline

g(c)=-3

for at least one

c

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4

Posted 3 months ago

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

-5

-2

\newline

g(c)=0

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-1

3

\newline

g(c)=-3

for at least one

c

-1

3

Posted 3 months ago

Question

f

be a continuous function on the closed interval

[-2,1]

f(-2)=3

f(1)=6

\newline

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

\newline

1

\newline

f(c)=0

for at least one

c

-2

1

\newline

f(c)=0

for at least one

c

3

6

\newline

f(c)=4

for at least one

c

-2

1

\newline

f(c)=4

for at least one

c

3

6

Posted 3 months ago

Question

f

be a continuous function on the closed interval

[-5,0]

f(-5)=0

f(0)=5

\newline

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

\newline

1

\newline

f(c)=-2

for at least one

c

-5

0

\newline

f(c)=2

for at least one

c

0

5

\newline

f(c)=2

for at least one

c

-5

0

\newline

f(c)=-2

for at least one

c

0

5

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Question

f

be a continuous function on the closed interval

[1,5]

f(1)=1

f(5)=-3

\newline

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

\newline

1

\newline

f(c)=2

for at least one

c

1

5

\newline

f(c)=-2

for at least one

c

-3

1

\newline

f(c)=-2

for at least one

c

1

5

\newline

f(c)=2

for at least one

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Posted 3 months ago

Question

h(x)=\frac{1}{x}

\newline

Can we use the intermediate value theorem to say the equation

h(x)=0.5

has a solution where

-1 \leq x \leq 1

\newline

1

\newline

(A) No, since the function is not continuous on that interval.

\newline

0

5

h(-1)

h(1)

\newline

(C) Yes, both conditions for using the intermediate value theorem have been met.

Posted 3 months ago

Question

h

be a continuous function on the closed interval

[0,4]

h(0)=2

h(4)=-2

\newline

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

\newline

1

\newline

h(c)=3

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

-2

2

\newline

h(c)=3

for at least one

c

-2

2

Posted 3 months ago