# Calculus

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)

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1. is x^(1/5) defined and continuous in that interval?
(-1)^(1/5) = -1 etc, seems ok

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

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2. 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)

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