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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 g be a continuous function on the closed interval [1,4] [-1,4] , where g(1)=4 g(-1)=-4 and g(4)=1 g(4)=1 .\newlineWhich of the following is guaranteed by the Intermediate Value Theorem?\newlineChoose 11 answer:\newline(A) g(c)=3 g(c)=-3 for at least one c c between 1-1 and 44\newline(B) g(c)=3 g(c)=3 for at least one c c between 4-4 and 11\newline(C) g(c)=3 g(c)=-3 for at least one c c between 4-4 and 11\newline(D) g(c)=3 g(c)=3 for at least one c c between 1-1 and 44

Full solution

Q. Let g g be a continuous function on the closed interval [1,4] [-1,4] , where g(1)=4 g(-1)=-4 and g(4)=1 g(4)=1 .\newlineWhich of the following is guaranteed by the Intermediate Value Theorem?\newlineChoose 11 answer:\newline(A) g(c)=3 g(c)=-3 for at least one c c between 1-1 and 44\newline(B) g(c)=3 g(c)=3 for at least one c c between 4-4 and 11\newline(C) g(c)=3 g(c)=-3 for at least one c c between 4-4 and 11\newline(D) g(c)=3 g(c)=3 for at least one c c between 1-1 and 44
  1. IVT Explanation: The Intermediate Value Theorem states that if a function is continuous on a closed interval [a,b][a, b] and NN is any number between f(a)f(a) and f(b)f(b), then there exists at least one cc in the interval [a,b][a, b] such that f(c)=Nf(c) = N.
  2. Given Function and Intervals: We are given that gg is continuous on the closed interval [1,4][-1, 4], g(1)=4g(-1) = -4, and g(4)=1g(4) = 1. We need to find if there is a value cc in the interval [1,4][-1, 4] such that g(c)g(c) equals a certain value.
  3. Option (A) Analysis: Option (A) suggests that g(c)=3g(c) = -3 for at least one cc between 1-1 and 44. Since 3-3 is between g(1)=4g(-1) = -4 and g(4)=1g(4) = 1, by the Intermediate Value Theorem, there must be at least one cc in the interval [1,4][-1, 4] where g(c)=3g(c) = -3.
  4. Option (B) Analysis: Option (B) suggests that g(c)=3g(c) = 3 for at least one cc between 4-4 and 11. However, the interval [4,1][-4, 1] is not the interval we are considering for the function gg, which is defined on [1,4][-1, 4]. Therefore, this option is not relevant to the given information.
  5. Option (C) Analysis: Option (C) suggests that g(c)=3g(c) = -3 for at least one cc between 4-4 and 11. Similar to option (B), this interval is not the one we are considering for the function gg, which is defined on [1,4][-1, 4]. Therefore, this option is also not relevant to the given information.
  6. Option (D) Analysis: Option (D) suggests that g(c)=3g(c) = 3 for at least one cc between 1-1 and 44. Since 33 is not between g(1)=4g(-1) = -4 and g(4)=1g(4) = 1, the Intermediate Value Theorem does not guarantee that there is a value cc where g(c)=3g(c) = 3 in the interval [1,4][-1, 4].

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