Determine which consecutive integers do not have a real zero of f(x) = x^3 + 5x^2 - x - 6 between them.
a. (-6, -5)
b. (-5, -4)
c. (-2, -1)
d. (1, 2)

Answer with Step-by-step explanation:

A function f(x) is said to have a real root in an interval (a,b) if value of the function at a and b have opposite signs.

f(x)=x³+5x²-x-6

A. f(-6) = -36, f(-5) = -1

Hence, f(x) has no real root between (-6,-5)

B. f(-5) = -1, f(-4) = 14

Hence, f(x) has a root between them.

C. f(-2) = 8, f(-1) = -1

Hence, (x) has a root between them.

D. f(1) = -1, f(2) = 20

Hence, f(x) has a root between them.

Hence, correct option is:

a. (-6,-5)

i think it's eleven too, don't woosh me here, but according to the "experts" it's 2.

the proof starts from the peano postulates, which define the natural

numbers n. n is the smallest set satisfying these postulates:

  p1.   1 is in n.

  p2.   if x is in n, then its "successor" x' is in n.

  p3.   there is no x such that x' = 1.

  p4.   if x isn't 1, then there is a y in n such that y' = x.

  p5.   if s is a subset of n, 1 is in s, and the implication

      (x in s => x' in s) holds, then s = n.

then you have to define addition recursively:

  def: let a and b be in n. if b = 1, then define a + b = a'

      (using p1 and p2). if b isn't 1, then let c' = b, with c in n

      (using p4), and define a + b = (a + c)'.

then you have to define 2:

  def:   2 = 1'

2 is in n by p1, p2, and the definition of 2.

theorem:   1 + 1 = 2

proof: use the first part of the definition of + with a = b = 1.

      then 1 + 1 = 1' = 2   q.e.d.

note: there is an alternate formulation of the peano postulates which

replaces 1 with 0 in p1, p3, p4, and p5. then you have to change the

definition of addition to this:

  def: let a and b be in n. if b = 0, then define a + b = a.

      if b isn't 0, then let c' = b, with c in n, and define

      a + b = (a + c)'.

you also have to define 1 = 0', and 2 = 1'. then the proof of the

theorem above is a little different:

proof: use the second part of the definition of + first:

      1 + 1 = (1 + 0)'

      now use the first part of the definition of + on the sum in

      parentheses:   1 + 1 = (1)' = 1' = 2   q.e.d.