Which of the following functions does not satisfy the conditions of the Mean Value Theorem on the interval [-1, 1]?
a. 5th root of x
b. 2x arccosx
c. x/(x - 3)
d. sqrt(x + 1)
is x^(1/5) defined and continuous in that interval?
(-1)^(1/5) = -1 etc, seems ok
how about arc cos x
that looks fine
x/(x-3) woukld blow up at x = 3, but we do not go there so it is ok
but
Oh my, sqrt(-1) is imaginary! and is called i for want of a real number to use. It is a no, no
a is ok, since it is differentiable on (-1,1). It has a vertical tangent, bot only at the endpoint(s).
b is ok for the same reason
c is ok
d is real on the interval, since it's √(x+1), so it's ok too
Looks to me like they all work, since the MVT requires
1. f(x) is defined and continuous on the interval [a,b]
2. differentiable on (a,b)
To determine which of the given functions does not satisfy the conditions of the Mean Value Theorem on the interval [-1, 1], we need to check if the function is continuous on the interval and differentiable on the open interval (-1, 1).
a. The 5th root of x is a continuous function on the interval [-1, 1] and is also differentiable on the open interval (-1, 1). Therefore, it satisfies the conditions of the Mean Value Theorem.
b. The function 2x arccosx is continuous on the interval [-1, 1]. However, it is not differentiable for x = -1 or x = 1 (endpoints of the interval). Therefore, it does not satisfy the conditions of the Mean Value Theorem.
c. The function x/(x - 3) is not defined at x = 3, which means it is not continuous on the interval [-1, 1]. Therefore, it does not satisfy the conditions of the Mean Value Theorem.
d. The function sqrt(x + 1) is a continuous function on the interval [-1, 1] and is also differentiable on the open interval (-1, 1). Therefore, it satisfies the conditions of the Mean Value Theorem.
Therefore, the function that does not satisfy the conditions of the Mean Value Theorem on the interval [-1, 1] is option c. x/(x - 3).
To determine which of the given functions does not satisfy the conditions of the Mean Value Theorem on the interval [-1, 1], we need to check if each function is continuous on the interval and differentiable on the open interval (-1, 1).
Let's go through each of the functions and verify these conditions:
a. 5th root of x: This function is continuous and differentiable on the interval (-1, 1), and therefore it satisfies the conditions of the Mean Value Theorem.
b. 2x arccosx: This function is continuous and differentiable on the interval (-1, 1), and thus it also satisfies the conditions of the Mean Value Theorem.
c. x/(x - 3): This function is defined for all values of x except x = 3. It is continuous on the interval (-1, 1), but it is not differentiable at x = 3. Since the Mean Value Theorem requires differentiability on the open interval (-1, 1), this function does not satisfy the conditions of the theorem.
d. sqrt(x + 1): This function is continuous and differentiable on the interval (-1, 1), and hence it satisfies the conditions of the Mean Value Theorem.
Therefore, the function that does not satisfy the conditions of the Mean Value Theorem on the interval [-1, 1] is c. x/(x - 3).