5.2 topic questions

x 0 1 2 3 --------------- f(x) 0 4 7 6 Let f be a function with selected values given in the table above. Which of the following statements must be true? I. By the Intermediate Value Theorem, there is a value c in the interval (0,3) such that f(c)=2. II. By the Mean Value Theorem, there is a value c in the interval (0,3) such that f′(c)=2. III. By the Extreme Value Theorem, there is a value c in the interval [0,3] such that f(c)≤f(x) for all x in the interval [0,3]. A) None B) I only C) II only D) I, II, and III

A

x 0 1 2 3 -------------------- f(x) 15 14 12 9 Let f be a function with selected values given in the table above. Which of the following statements must be true? I. By the Intermediate Value Theorem, there is a value c in the interval (0,3) such that f(c)=10. II. By the Mean Value Theorem, there is a value c in the interval (0,3) such that f′(c)=−2. III. By the Extreme Value Theorem, there is a value c in the interval [0,3] such that f(c) ≤ f(x) for all x in the interval [0,3]. A) None B) I only C) II only D) I, II, and III

A

Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0,2π]? A) y=1/(1 + sin⁡x) B) y=1/(x^2 + π) C) y=(x^2 − 2πx + π^2)/(x − π) D) y=|x − π|/x − π

B

Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0,4] ? A) y = tanx B) y=tan^−1(x) C) y = (x^2 − 16)/(x^2 + x −20) D) y = 1/(e^x − 1)

B

Let g be the function given by g(x)=√(1+cosx). Which of the following statements could be false on the interval π/2 ≤ x ≤ 7π/4? A) By the Extreme Value Theorem, there is a value c such that g(c) ≤ g(x) for π/2 ≤ x ≤ 7π/4. B) By the Extreme Value Theorem, there is a value c such that g(c) ≥ g(x) for π/2 ≤ x ≤ 7π/4. C) By the Intermediate Value Theorem, there is a value c such that g(c) = (g(π/2) + g(7π/4))/2. D) By the Mean Value Theorem, there is a value c such that g'(c) = (g(7π/4) − g(π/2))/(7π/4 − π/2).

D

Let g be the function given by g(x)=√(1−sin(x)^2). Which of the following statements could be false on the interval 0 ≤ x≤ π? A) By the Extreme Value Theorem, there is a value c such that g(c) ≤ g(x) for 0 ≤ x ≤ π. B) By the Extreme Value Theorem, there is a value c such that g(c) ≥ g(x) for 0 ≤ x ≤ π. C) By the Intermediate Value Theorem, there is a value c such that g(c) = (g(0)+g(π))/2. D) By the Mean Value Theorem, there is a value c such that g′(c) = (g(π)−g(0))/(π−0).

D