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

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

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Intermediate Value Theorem

Full solution

Q. Let

f

be a continuous function on the closed interval

[-5,0]

, where

f(-5)=0

and

f(0)=5

.

\newline

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

\newline

Choose

1

answer:

\newline

(A)

f(c)=2

for at least one

c

between

-5

and

0

\newline

(B)

f(c)=-2

for at least one

c

between

0

and

5

\newline

(C)

f(c)=2

for at least one

c

between

0

and

5

\newline

(D)

f(c)=-2

for at least one

c

between

-5

and

0

Apply Intermediate Value Theorem: 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 $c$ in the interval $[a, b]$ such that $f(c) = N$ . We need to apply this theorem to the given function $f$ on the interval $[-5, 0]$ .
Given Function Values: We are given that $f(-5) = 0$ and $f(0) = 5$ . This means that the function $f$ takes on all values between $0$ and $5$ on the interval $[-5, 0]$ .
Option (A) Analysis: Option (A) asks if $f(c) = 2$ for at least one $c$ between $-5$ and $0$ . Since $2$ is between $0$ and $5$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $[-5, 0]$ such that $f(c) = 2$ .
Option (B) Analysis: Option (B) asks if $f(c) = -2$ for at least one $c$ between $0$ and $5$ . This is not applicable because the interval given in the problem is $[-5, 0]$ , not $[0, 5]$ .
Option (C) Analysis: Option (C) asks if $f(c) = 2$ for at least one $c$ between $0$ and $5$ . Again, this is not applicable because the interval given in the problem is $[-5, 0]$ , not $[0, 5]$ .
Option (D) Analysis: Option (D) asks if $f(c) = -2$ for at least one $c$ between $-5$ and $0$ . Since $-2$ is not between the values of $f(-5) = 0$ and $f(0) = 5$ , the Intermediate Value Theorem does not guarantee that there is a $c$ in the interval $[-5, 0]$ such that $f(c) = -2$ .

More problems from Intermediate Value Theorem

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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

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for at least one

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h

be a continuous function on the closed interval

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\newline

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Question

g

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\newline

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\newline

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3

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6

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Question

g

be a continuous function on the closed interval

[-1,4]

g(-1)=-4

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\newline

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

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1

\newline

g(c)=3

for at least one

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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

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\newline

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

\newline

1

\newline

g(c)=-3

for at least one

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-2

\newline

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g(c)=0

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Question

f

be a continuous function on the closed interval

[-2,1]

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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

-2

1

\newline

f(c)=0

for at least one

c

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6

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1

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f

be a continuous function on the closed interval

[-5,0]

f(-5)=0

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\newline

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

\newline

1

\newline

f(c)=-2

for at least one

c

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0

\newline

f(c)=2

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Question

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\newline

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f(c)=2

for at least one

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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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Question

h

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\newline

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1

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for at least one

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