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

The Intermediate Value Theorem: The Intermediate Value Theorem states that if f is a continuous function 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.Given Function Values: We are given that f(1) = 1 and f(5) = -3. This means that the function f changes from a positive value to a negative value as x increases from 1 to 5.Application of Theorem: Since f is continuous on [1, 5], by the Intermediate Value Theorem, for any value N between 1 and -3, there must be some c in the interval [1, 5] such that f(c) = N.Elimination of Incorrect Options: Looking at the options, we can eliminate (A) and (D) because 2 is not between 1 and -3, so there is no guarantee that there is a c such that f(c) = 2.Correct Option Explanation: Option (B) is incorrect because it refers to a c between -3 and 1, but our interval is from 1 to 5.Option (C) states that f(c) = -2 for at least one c between 1 and 5. Since -2 is between 1 and -3, the Intermediate Value Theorem guarantees that there is at least one c in the interval [1, 5] such that f(c) = -2.

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

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

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Intermediate Value Theorem

Full solution

Q. Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1

and

f(5)=-3

\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

1

and

5

\newline

(B)

f(c)=-2

for at least one

c

between

-3

and

1

\newline

(C)

f(c)=-2

for at least one

c

between

1

and

5

\newline

(D)

f(c)=2

for at least one

c

between

-3

and

1

The Intermediate Value Theorem: The Intermediate Value Theorem states that if $f$ is a continuous function 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 $f(c) = N$ .
Given Function Values: We are given that $f(1) = 1$ and $f(5) = -3$ . This means that the function $f$ changes from a positive value to a negative value as $x$ increases from $1$ to $5$ .
Application of Theorem: Since $f$ is continuous on $[1, 5]$ , by the Intermediate Value Theorem, for any value $N$ between $1$ and $-3$ , there must be some $c$ in the interval $[1, 5]$ such that $f(c) = N$ .
Elimination of Incorrect Options: Looking at the options, we can eliminate $(A)$ and $(D)$ because $2$ is not between $1$ and $-3$ , so there is no guarantee that there is a $c$ such that $f(c) = 2$ .
Correct Option Explanation: Option (B) is incorrect because it refers to a $c$ between $-3$ and $1$ , but our interval is from $1$ to $5$ .
Correct Option Explanation: Option (B) is incorrect because it refers to a $c$ between $-3$ and $1$ , but our interval is from $1$ to $5$ .Option (C) states that $f(c) = -2$ for at least one $c$ between $1$ and $5$ . Since $-2$ is between $1$ and $-3$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $-3$ $3$ such that $f(c) = -2$ .

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1