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 -3 and 3 (B) g(c)=-2 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)=5 for at least one c between 0 and 6

IVT Explanation: 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 [-3, 3], g(-3) = 0, and g(3) = 6.Option (A) Analysis: We need to determine which of the given options is guaranteed by the Intermediate Value Theorem. Let's analyze each option: (A) g(c) = -2 for at least one c between -3 and 3. Since g(-3) = 0 and the values of g on the interval [-3, 3] are between 0 and 6, -2 is not between g(-3) and g(3). Therefore, the Intermediate Value Theorem does not guarantee that there is a c such that g(c) = -2.Option (B) Analysis: (B) g(c) = -2 for at least one c between 0 and 6. This option is not relevant because the interval for c should be between -3 and 3, not 0 and 6. Additionally, -2 is not between g(-3) and g(3).Option (C) Analysis: (C) g(c) = 5 for at least one c between -3 and 3. Since g(-3) = 0 and g(3) = 6, and 5 is between 0 and 6, the Intermediate Value Theorem guarantees that there is at least one c in the interval [-3, 3] such that g(c) = 5.Option (D) Analysis: (D) g(c) = 5 for at least one c between 0 and 6. This option is not relevant because the interval for c should be between -3 and 3, not 0 and 6. However, since 5 is between g(-3) and g(3), there is a guarantee that there is a c such that g(c) = 5, but the interval for c is incorrect in this option.

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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 -3 and 3
(B)
g(c)=-2 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)=5 for at least one
c between 0 and 6

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 $-3$ and $3$ $\newline$ (B) $g(c)=-2$ 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) $g(c)=5$ for at least one $c$ between $0$ and $6$

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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

-3

and

3

\newline

(B)

g(c)=-2

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)=5

for at least one

c

between

0

and

6

IVT Explanation: 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 $[-3, 3]$ , $g(-3) = 0$ , and $N$ $0$ .
Option (A) Analysis: We need to determine which of the given options is guaranteed by the Intermediate Value Theorem. Let's analyze each option: $\newline$ (A) $g(c) = -2$ for at least one $c$ between $-3$ and $3$ . Since $g(-3) = 0$ and the values of $g$ on the interval $[-3, 3]$ are between $0$ and $6$ , $-2$ is not between $c$ $0$ and $c$ $1$ . Therefore, the Intermediate Value Theorem does not guarantee that there is a $c$ such that $g(c) = -2$ .
Option (B) Analysis: (B) $g(c) = -2$ for at least one $c$ between $0$ and $6$ . This option is not relevant because the interval for $c$ should be between $-3$ and $3$ , not $0$ and $6$ . Additionally, $-2$ is not between $c$ $0$ and $c$ $1$ .
Option (C) Analysis: (C) $g(c) = 5$ for at least one $c$ between $-3$ and $3$ . Since $g(-3) = 0$ and $g(3) = 6$ , and $5$ is between $0$ and $6$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $c$ $0$ such that $g(c) = 5$ .
Option (D) Analysis: (D) $g(c) = 5$ for at least one $c$ between $0$ and $6$ . This option is not relevant because the interval for $c$ should be between $-3$ and $3$ , not $0$ and $6$ . However, since $5$ is between $c$ $0$ and $c$ $1$ , there is a guarantee that there is a $c$ such that $g(c) = 5$ , but the interval for $c$ is incorrect in this option.

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1