Mean Value and Rolle's Theorem

Let f be the function defined by f(x) = x³+x-4. What is the value of c for which the instantaneous rate of change of f at x=c is the same as the average rate of change of f over [-1, 3]. 

• 0
• 1
• √(2/3)
• √(7/3)

If f(x)=x³, at what point on the interval 0<x<3 is the tangent to the curve parallel to the secant line for this interval? 

• 3
• √3
• √3/3
• 1/3

If f(x)=sin(x/2), at what point on the interval π/2<x<3π/2 is the tangent to the curve parallel to the secant line for this interval? 

• 2π/3
• 3π/4
• 5π/6
• π

On the closed interval [-2,3], the absolute maximum value(s) of f(x) = x³-3x²+12 occurs at x = 

• 0 and 3
• 0
• 0 and 2
• -2 and 0

Suppose that f is differentiable for all reals. For x in the interval (-5, 0), suppose that f'(x)<0. For x in the interval (0, 5), suppose that f'(x) >0. Then at x = 0 it must be true that f has 

• an inflection point
• an absolute maximum
• an absolute minimum
• a relative minimum

Suppose that f is differentiable for all reals. For x in the interval (-∞, 4), suppose that f'(x)>0. For x in the interval (4, ∞), suppose that f'(x) <0. Then at x = 4 it must be true that f has 

• no extrema
• a relative minimum
• an absolute maximum
• an absolute minimum

Let g be a continuous function of the closed interval [-1, 3]. If g(-1)=-10 and g(3) = 6, which of the following are guaranteed? 

• g(0) = 0
• g'(c)= 0 for some c in (-1,3)
• g'(c)= 4 for some c in (-1, 3)
• g(c)= 4 for some c in (-1, 3)

The function h is continuous on the closed interval [-2, 5] and differentiable on the open interval (-2, 5) . If f (-2) = 3 and f (5) =-11, which of the following statements could be false?

• There exists a c on the interval (-2, 5) such that f'(c) = 0. 
• There is an absolute maximum value on the interval [-2, 5]
• There exists a con on the interval (-2, 5) such that f(c) = 0. 
• There exists a c on the interval (-2, 5) such that f'(c) = -2. 

Let g be a function that is differentiable over the interval (2, 9) . Giveng(3) = 5 , g(6) = -2 , and g(8) = 5 , which of the following must be true?
I. g has at least one horizontal tangent line.
II. g has at least 2 zeros.
III. For some c in the interval (3, 6), f'(c) = -7/3. 

• II only
• II and III
• I and II
• I, II, and III

If f(x) = x³ + 1 then there exists a number c in the interval (0, 1) that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c ?

• -√(1/3)
• √(1/3)
• 1
• 2