I would appreciate it if anyone could shed some light as to the answer/procedure to reach an answer for the following problems. Anything to get me started will do.

I

If c > 0 and f(x) = e^(x) - cx for all real number x, then the minimum value of f is

a) f(c)

b) f(e^c)

c) f(1/c)

ans d) f(log c)

e) nonexistent

y' = e^x - c, when y' = 0 c = e^x

y'' = e^x

Don't see a way to discern the min value...

II

Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11 then f(15/2) is

a) -11

b) 0

ans c) 11

d) 33/2

e) not uniquely determined

Don't know how to get this one started

III Other than trying out the answers, is there a quicker way?

Let x and y be positive integers such that 3x + 7y is divisible by 11. Which of the following must also be divisible by 11?

a) 4x + 6y

b) x + y + 5

c) 9x + 4y

ans d) 4x - 9y

e) x + y - 1

IV This one seems common...

If a polynomial f(x) over the real numbers has the complex numbers 2 + i and 1 - i as roots, then f(x) could be

a) x^4 + 6X^3 + 10

b) x^4 + 7x^2 +10

c) x^3 - x^2 + 4x +1

d) x^3 + 5x^2 + 4x +1

ans e) x^4 - 6x^3 + 15x^2 - 18x +10

I realize that these roots occur in complex conjugate pairs... Checking the answers seems like too much work...

V

In a game two players take turns tossing a fair coin; the winner is the first one toss a head. The probability that the player who makes the first toss wins the game is

a) 1/4

b) 1/3

c) 1/2

ans d) 2/3

e) 3/4

This one might be about the wording. I don't see why the answer is 2/3

VI

Let x(sub 1) = 1 and x(sub n + 1) = sqrt(3 + 2*n(sub n)) for all positive integers n. If it is assumed that {x(sub n)} converges, then lim x -> infiniti x(sub n) =

a) -1

b) 0

c) sqrt(5)

d) e

ans e) 3

I get this somewhat nasty nested function of square roots and basically I do not see how it may simplify...