Help with the following problems greatly appreciated
Posted: Tue Oct 30, 2007 11:33 pm
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...
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...