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

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 g(3) = 6.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.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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 $3$

View step-by-step help

Home

Math Problems

Calculus

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

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1