Let f be a continuous function on the closed interval [-5,0], where f(-5)=0 and f(0)=5. Which of the following is guaranteed by the Intermediate Value Theorem? Choose 1 answer: (A) f(c)=-2 for at least one c between -5 and 0 (B) f(c)=2 for at least one c between 0 and 5 (C) f(c)=2 for at least one c between -5 and 0 (D) f(c)=-2 for at least one c between 0 and 5

Intermediate Value Theorem: The Intermediate Value Theorem states that if a function f 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 Values: We are given that f(-5) = 0 and f(0) = 5, which means that the function f takes on the value 0 at x = -5 and the value 5 at x = 0.Determine Value of c: We need to determine if there is a value c in the interval [-5, 0] for which f(c) is either -2 or 2, according to the options provided.Option (A) Analysis: Option (A) suggests that f(c) = -2 for some c between -5 and 0. However, since f(-5) = 0 and f(0) = 5, the value -2 is not between f(-5) and f(0). Therefore, the Intermediate Value Theorem does not guarantee a c such that f(c) = -2 in the interval [-5, 0].Option (B) Analysis: Option (B) suggests that f(c) = 2 for some c between 0 and 5. This option is not relevant because our interval is [-5, 0], not [0, 5].Option (C) Analysis: Option (C) suggests that f(c) = 2 for some c between -5 and 0. Since 2 is between f(-5) = 0 and f(0) = 5, the Intermediate Value Theorem guarantees that there is at least one c in the interval [-5, 0] such that f(c) = 2.Option (D) Analysis: Option (D) suggests that f(c) = -2 for some c between 0 and 5. Again, this option is not relevant because our interval is [-5, 0], not [0, 5].

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Let
f be a continuous function on the closed interval
[-5,0], where
f(-5)=0 and
f(0)=5.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
f(c)=-2 for at least one
c between -5 and 0
(B)
f(c)=2 for at least one
c between 0 and 5
(C)
f(c)=2 for at least one
c between -5 and 0
(D)
f(c)=-2 for at least one
c between 0 and 5

Let $f$ be a continuous function on the closed interval $[-5,0]$ , where $f(-5)=0$ and $f(0)=5$ . $\newline$ Which of the following is guaranteed by the Intermediate Value Theorem? $\newline$ Choose $1$ answer: $\newline$ (A) $f(c)=-2$ for at least one $c$ between $-5$ and $0$ $\newline$ (B) $f(c)=2$ for at least one $c$ between $0$ and $5$ $\newline$ (C) $f(c)=2$ for at least one $c$ between $-5$ and $0$ $\newline$ (D) $f(c)=-2$ for at least one $c$ between $0$ and $5$

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Calculus

Intermediate Value Theorem

Full solution

Q. Let

f

be a continuous function on the closed interval

[-5,0]

, where

f(-5)=0

and

f(0)=5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

Choose

1

answer:

\newline

(A)

f(c)=-2

for at least one

c

between

-5

and

0

\newline

(B)

f(c)=2

for at least one

c

between

0

and

5

\newline

(C)

f(c)=2

for at least one

c

between

-5

and

0

\newline

(D)

f(c)=-2

for at least one

c

between

0

and

5

Intermediate Value Theorem: The Intermediate Value Theorem states that if a function $f$ 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 Values: We are given that $f(-5) = 0$ and $f(0) = 5$ , which means that the function $f$ takes on the value $0$ at $x = -5$ and the value $5$ at $x = 0$ .
Determine Value of $c$ : We need to determine if there is a value $c$ in the interval $[-5, 0]$ for which $f(c)$ is either $-2$ or $2$ , according to the options provided.
Option (A) Analysis: Option (A) suggests that $f(c) = -2$ for some $c$ between $-5$ and $0$ . However, since $f(-5) = 0$ and $f(0) = 5$ , the value $-2$ is not between $f(-5)$ and $f(0)$ . Therefore, the Intermediate Value Theorem does not guarantee a $c$ such that $f(c) = -2$ in the interval $c$ $1$ .
Option (B) Analysis: Option (B) suggests that $f(c) = 2$ for some $c$ between $0$ and $5$ . This option is not relevant because our interval is $[-5, 0]$ , not $[0, 5]$ .
Option (C) Analysis: Option (C) suggests that $f(c) = 2$ for some $c$ between $-5$ and $0$ . Since $2$ is between $f(-5) = 0$ and $f(0) = 5$ , the Intermediate Value Theorem guarantees that there is at least one $c$ in the interval $[-5, 0]$ such that $f(c) = 2$ .
Option (D) Analysis: Option (D) suggests that $f(c) = -2$ for some $c$ between $0$ and $5$ . Again, this option is not relevant because our interval is $[-5, 0]$ , not $[0, 5]$ .

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1