2019 international practice exam ab mcq, part b - calculator required the table above gives values of a…

2019 international practice exam ab mcq, part b - calculator required the table above gives values of a continuous function f at selected values of x. 9 mark for review based on the information in the table, which of the following statements must be true? a f has at most one zero. b f has a relative maximum at x = 2. c there exists a value c, where -5 < c < 2, such that f(c)=4. d there exists a value c, where -5 < c < 2, such that f(c)=2.

2019 international practice exam ab mcq, part b - calculator required the table above gives values of a continuous function f at selected values of x. 9 mark for review based on the information in the table, which of the following statements must be true? a f has at most one zero. b f has a relative maximum at x = 2. c there exists a value c, where -5 < c < 2, such that f(c)=4. d there exists a value c, where -5 < c < 2, such that f(c)=2.

Answer

Explanation:

Step1: Recall Intermediate - Value Theorem

The Intermediate - Value Theorem states that if (y = f(x)) is continuous on the closed interval ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c) in the open interval ((a,b)) such that (f(c)=k).

Step2: Analyze the given function values

We have (f(-5)=-9), (f(0) = 1), and (f(2)=5). The function (y = f(x)) is continuous on the interval ([-5,2]). The number (4) is between (1) and (5) (since (1<4<5)).

Step3: Apply the Intermediate - Value Theorem

By the Intermediate - Value Theorem, since (f(x)) is continuous on ([-5,2]) and (4) is between (f(0) = 1) and (f(2)=5), there exists a value (c) in the open interval ((-5,2)) such that (f(c)=4).

Step4: Analyze other options

  • Option A: Just from the three - point table, we cannot be sure that (f) has at most one zero. There could be more zeros between the non - listed (x) values.
  • Option B: We have no information about the derivative of (f) or the behavior of (f) around (x = 2) to conclude that it has a relative maximum at (x = 2).
  • Option D: We have no information about the derivative of (f) to conclude that there exists a (c\in(-5,2)) such that (f^{\prime}(c)=2).

Answer:

C. There exists a value (c), where (-5 < c < 2), such that (f(c)=4)