Multiple-Choice Questions

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1. Let f be a differentiable odd function defined on R. (That is f(-x) = -f(x) for all x in R.) Let a be a positive number. How many solution(s) could the equation af'(x) = f(a) have?
a) 0
b) 1
c) 2
d) 3

2. Let f be a quadratic function defined on the interval [a,b] with 0 < a < b. Which one of the followings is the value of c in (a,b) such that f(b)-f(a) = f'(c)(b-a) ?
a) (a+b)/2
b) (ab)½
c) 2ab/(a+b)
d) Cannot be determined

3. The Mean Value Theorem is applied to the function f(x) = x3 + qx2 + 5x - 6 on the interval [0,2]. Suppose that the number c determined by the theorem is equal to 2. Which one of the followings is the value of q ?
a) -1
b) -2
c) -3
d) -4

4. Which of the following functions satisfy the conditions of the Mean value Theorem on their domains ?
(I) f(x) = x¾ for all x in [-1,1].
(II) g(x) = |x-1| for all x in [-1,1].
(III) h(x) = x/(1-x2) for all x in (-1,1).
(IV) k(x) = 1-|x|3 for all x in [-1,½].
a) (I) and (III)
b) (I) and (IV)
c) (II) and (III)
d) (II) and (IV)

5. Let g(x) be a differentiable function defined on R and f(x) = g(x)sinx. How many real solutions does the equation g(x)cosx + g'(x)sinx = 0 have ?
a) 0
b) 1
c) 2
d) Infinitely many


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