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 -5 and -2 (B) g(c)=0 for at least one c between -5 and -2 (C) g(c)=0 for at least one c between -1 and 3 (D) g(c)=-3 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 g(3) = -5.Analysis of Options: We need to determine which statement is guaranteed by the Intermediate Value Theorem. Let's analyze each option: (A) g(c) = -3 for at least one c between -5 and -2: This statement is incorrect because -5 and -2 are values of g, not values of c.Correct Statement: (B) g(c) = 0 for at least one c between -5 and -2: This statement is incorrect because it refers to values of c between -5 and -2, which are not in the domain of the function g, but rather in its range.Application of Theorem: (C) g(c) = 0 for at least one c between -1 and 3: This statement is a possible application of the Intermediate Value Theorem since 0 is between the values of g(-1) = -2 and g(3) = -5, and the interval [-1, 3] is the domain over which g is continuous.(D) g(c) = -3 for at least one c between -1 and 3: This statement is correct because -3 is a value between g(-1) = -2 and g(3) = -5, and according to the Intermediate Value Theorem, since g is continuous on [-1, 3], there must be at least one c in the interval [-1, 3] such that g(c) = -3.

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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 -5 and -2
(B)
g(c)=0 for at least one
c between -5 and -2
(C)
g(c)=0 for at least one
c between -1 and 3
(D)
g(c)=-3 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 $-5$ and $-2$ $\newline$ (B) $g(c)=0$ for at least one $c$ between $-5$ and $-2$ $\newline$ (C) $g(c)=0$ for at least one $c$ between $-1$ and $3$ $\newline$ (D) $g(c)=-3$ for at least one $c$ between $-1$ and $3$

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Math Problems

Calculus

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

-5

and

-2

\newline

(B)

g(c)=0

for at least one

c

between

-5

and

-2

\newline

(C)

g(c)=0

for at least one

c

between

-1

and

3

\newline

(D)

g(c)=-3

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 $N$ $0$ .
Analysis of Options: We need to determine which statement is guaranteed by the Intermediate Value Theorem. Let's analyze each option: $\newline$ (A) $g(c) = -3$ for at least one $c$ between $-5$ and $-2$ : This statement is incorrect because $-5$ and $-2$ are values of $g$ , not values of $c$ .
Correct Statement: (B) $g(c) = 0$ for at least one $c$ between $-5$ and $-2$ : This statement is incorrect because it refers to values of $c$ between $-5$ and $-2$ , which are not in the domain of the function $g$ , but rather in its range.
Application of Theorem: (C) $g(c) = 0$ for at least one $c$ between $-1$ and $3$ : This statement is a possible application of the Intermediate Value Theorem since $0$ is between the values of $g(-1) = -2$ and $g(3) = -5$ , and the interval $[-1, 3]$ is the domain over which $g$ is continuous.
Application of Theorem: (C) $g(c) = 0$ for at least one $c$ between $-1$ and $3$ : This statement is a possible application of the Intermediate Value Theorem since $0$ is between the values of $g(-1) = -2$ and $g(3) = -5$ , and the interval $[-1, 3]$ is the domain over which $g$ is continuous.(D) $g(c) = -3$ for at least one $c$ between $-1$ and $3$ : This statement is correct because $c$ $3$ is a value between $g(-1) = -2$ and $g(3) = -5$ , and according to the Intermediate Value Theorem, since $g$ is continuous on $[-1, 3]$ , there must be at least one $c$ in the interval $[-1, 3]$ such that $g(c) = -3$ .

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1