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 -1 and 4 (B) g(c)=-3 for at least one c between -4 and 1 (C) g(c)=3 for at least one c between -4 and 1 (D) g(c)=-3 for at least one c between -1 and 4

Apply Intermediate Value Theorem: The Intermediate Value Theorem states that if a function g is continuous on a closed interval [a, b] and N is any number between g(a) and g(b), then there exists at least one c in the interval [a, b] such that g(c) = N. We need to apply this theorem to the function g given its values at -1 and 4.Check Endpoint Values: First, we check the value of g at the endpoints of the interval. We have g(-1) = -4 and g(4) = 1. This means that the function g takes on all values between -4 and 1 on the interval [-1, 4].Examine Answer Choices: Now, we examine each answer choice to see which value is guaranteed to be taken by the function g on the interval [-1, 4] based on the Intermediate Value Theorem. (A) g(c)=3 for at least one c between -1 and 4 (B) g(c)=-3 for at least one c between -4 and 1 (C) g(c)=3 for at least one c between -4 and 1 (D) g(c)=-3 for at least one c between -1 and 4 We can immediately eliminate choices (A) and (C) because the value 3 is not between -4 and 1, the range of g on the interval [-1, 4].Eliminate Incorrect Choices: Next, we look at choice (B), which is incorrect because it refers to an interval [-4, 1] for the value of c, which is not the interval we are considering for the function g. We are only considering the interval [-1, 4] for the function g.Consider Correct Choice: Finally, we consider choice (D). Since -3 is a value between -4 and 1, and g is continuous on [-1, 4], the Intermediate Value Theorem guarantees that there is at least one c in the interval [-1, 4] such that g(c) = -3. Therefore, choice (D) is the correct answer.

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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 -1 and 4
(B)
g(c)=-3 for at least one
c between -4 and 1
(C)
g(c)=3 for at least one
c between -4 and 1
(D)
g(c)=-3 for at least one
c between -1 and 4

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 $-1$ and $4$ $\newline$ (B) $g(c)=-3$ for at least one $c$ between $-4$ and $1$ $\newline$ (C) $g(c)=3$ for at least one $c$ between $-4$ and $1$ $\newline$ (D) $g(c)=-3$ for at least one $c$ between $-1$ and $4$

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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

-1

and

4

\newline

(B)

g(c)=-3

for at least one

c

between

-4

and

1

\newline

(C)

g(c)=3

for at least one

c

between

-4

and

1

\newline

(D)

g(c)=-3

for at least one

c

between

-1

and

4

Apply Intermediate Value Theorem: The Intermediate Value Theorem states that if a function $g$ is continuous on a closed interval $[a, b]$ and $N$ is any number between $g(a)$ and $g(b)$ , then there exists at least one $c$ in the interval $[a, b]$ such that $g(c) = N$ . We need to apply this theorem to the function $g$ given its values at $-1$ and $[a, b]$ $0$ .
Check Endpoint Values: First, we check the value of $g$ at the endpoints of the interval. We have $g(-1) = -4$ and $g(4) = 1$ . This means that the function $g$ takes on all values between $-4$ and $1$ on the interval $[-1, 4]$ .
Examine Answer Choices: Now, we examine each answer choice to see which value is guaranteed to be taken by the function $g$ on the interval $[-1, 4]$ based on the Intermediate Value Theorem. $\newline$ (A) $g(c)=3$ for at least one $c$ between $-1$ and $4$ $\newline$ (B) $g(c)=-3$ for at least one $c$ between $-4$ and $1$ $\newline$ (C) $g(c)=3$ for at least one $c$ between $-4$ and $1$ $\newline$ (D) $g(c)=-3$ for at least one $c$ between $-1$ and $4$ $\newline$ We can immediately eliminate choices (A) and (C) because the value $[-1, 4]$ $8$ is not between $-4$ and $1$ , the range of $g$ on the interval $[-1, 4]$ .
Eliminate Incorrect Choices: Next, we look at choice (B), which is incorrect because it refers to an interval $[-4, 1]$ for the value of $c$ , which is not the interval we are considering for the function $g$ . We are only considering the interval $[-1, 4]$ for the function $g$ .
Consider Correct Choice: Finally, we consider choice (D). Since $-3$ is a value between $-4$ and $1$ , and $g$ is continuous on $[-1, 4]$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $[-1, 4]$ such that $g(c) = -3$ . Therefore, choice (D) is the correct answer.

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1