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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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Math Problems

Algebra 2

Identify discrete and continuous random variables

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

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 $f(c) = N$ .
Given Function and Intervals: We are given that $g$ is continuous on the closed interval $[-1, 4]$ , $g(-1) = -4$ , and $g(4) = 1$ . We need to find if there is a value $c$ in the interval $[-1, 4]$ such that $g(c)$ equals a certain value.
Option (A) Analysis: Option (A) suggests that $g(c) = -3$ for at least one $c$ between $-1$ and $4$ . Since $-3$ is between $g(-1) = -4$ and $g(4) = 1$ , by the Intermediate Value Theorem, there must be at least one $c$ in the interval $[-1, 4]$ where $g(c) = -3$ .
Option (B) Analysis: Option (B) suggests that $g(c) = 3$ for at least one $c$ between $-4$ and $1$ . However, the interval $[-4, 1]$ is not the interval we are considering for the function $g$ , which is defined on $[-1, 4]$ . Therefore, this option is not relevant to the given information.
Option (C) Analysis: Option (C) suggests that $g(c) = -3$ for at least one $c$ between $-4$ and $1$ . Similar to option (B), this interval is not the one we are considering for the function $g$ , which is defined on $[-1, 4]$ . Therefore, this option is also not relevant to the given information.
Option (D) Analysis: Option (D) suggests that $g(c) = 3$ for at least one $c$ between $-1$ and $4$ . Since $3$ is not between $g(-1) = -4$ and $g(4) = 1$ , the Intermediate Value Theorem does not guarantee that there is a value $c$ where $g(c) = 3$ in the interval $[-1, 4]$ .