Given f(x) = -1/x, find all c in the interval [-3, -½] that satisfies the Mean Value Theorem.
A. c= -sqrt(3/2)
B. c= +or- sqrt(3/2)
C. The Mean Value Theorem doesn’t apply because f is not continuous at x=0
D. The Mean Value Theorem doesn’t apply because f(-½) does not equal f(-3)
E. none of these
I think I got it. Is it B?
f' = 1/x^2
at x = -3, f = 1/3
at x = -1/2, f = 2
slope = (2 - 1/3)/(-1/2 + 3)
= (5/3)/(5/2) = 2/3
where is the derivative = 2/3 ?
1/x^2 = 2/3
x^2 = 3/2
x = +/- sqrt (3/2)
yes B
I believe the answer is (A), given the interval [-3,-½]
To determine which options satisfy the Mean Value Theorem for the given function and interval, we need to check if two conditions are met:
1. The function f(x) must be continuous on the closed interval [-3, -½].
2. The function f(x) must be differentiable on the open interval (-3, -½).
Let's go through each option and check if it satisfies the conditions of the Mean Value Theorem:
A. c= -√(3/2): This option can be immediately ruled out since it doesn't fall within the given interval [-3, -½].
B. c= ±√(3/2): For this option, we need to check if the function f(x) is continuous at c and differentiable on the open interval (-3, -½).
At c= ±√(3/2), the function f(x) is continuous since it is defined for all x ≠ 0. However, the function f(x) is not defined at x = 0, which falls within the interval (-3, -½). Therefore, the function is not differentiable and this option doesn't satisfy the conditions of the Mean Value Theorem.
C. The Mean Value Theorem doesn't apply because f is not continuous at x=0: This option suggests that since the function f(x) is not continuous at x = 0, the Mean Value Theorem doesn't apply. However, this is incorrect because the interval given is [-3, -½], and x=0 is not included in this interval. Therefore, this option can be ruled out.
D. The Mean Value Theorem doesn't apply because f(-½) does not equal f(-3): This option suggests that since the function f(-½) does not equal f(-3), the Mean Value Theorem doesn't apply. However, the Mean Value Theorem does not require the function values at the endpoints to be equal. Therefore, this option can be ruled out.
E. None of these: Since none of the previous options satisfy the conditions of the Mean Value Theorem, the correct answer is none of these.
Therefore, the correct answer is E. None of these.