INTERMEDIATE AND EXTREME VALUES

• Upper bound


  A number ! such that ≤ ! for all x in the set
• Lower bound


  A number ' such that x ≥ ' for all x in the set
• Upper bound of a function


  The set {y ∈ R|y = f(x) for some x ∈ X} has an upper bound
• Lower bound of a function

  The set {y ∈ R|y = f(x) for some x ∈ X} has a lower bound
• Supremum


  The least upper bound of a set
• Infimum


  The greatest lower bound of a set
• If a nonempty set of real numbers has a lower bound, then it has a greatest lower bound or infimum. Equivalently, if it has an upper bound, then it has a least upper bound or supremum.
• Intermediate Value Theorem


  If a function f is continuous on a closed interval [a,b] and f(a) ≠ f(b), then for any number E between f(a) and f(b), there exists a point c in [a,b] such that f(c) = E.
• Proof of Intermediate Value Theorem (Case 1)

  1. Define the set S = {x ∈ [a,b]|f(x) < E}

  2. Note S is nonempty since a ∈ S

  3. The supremum c = sup S exists

  4. Show f(c) = E
• Proof of Intermediate Value Theorem (Case 2)

  1. Define g(x) = -f(x) on [a,b]

  2. Since g(a) > g(b), by Case 1 there exists c ∈ [a,b] such that g(c) = -E

  3. Therefore, f(c) = E
• Corollary 1.1


  If a function f is continuous on [a,b] and f(a)·f(b) < 0, then there is at least one number c in (a,b) such that f(c) = 0.
• Bisection Method


  1. Set a margin of error ε

  2. Define x1 = a if f(a) < 0, else x1 = b

  3. Define x2 = a if x1 = b, else x2 = b

  4. Define xm = (x1 + x2)/2

  5. If |f(xm)| ≤ ε, take x = xm and end. Else, if f(xm) < 0, let x1 = xm. Else, let x2 = xm and go back to step 3.
• Absolute maximum


  A function f attains its absolute maximum at x0 if f(x) ≤ f(x0) for all x in the domain
• Absolute minimum


  A function f attains its absolute minimum at x0 if f(x) ≥ f(x0) for all x in the domain
• Local maximum


  A function f attains a local maximum at x0 if f(x) ≤ f(x0) for all x in (f(x0) - δ, f(x0) + δ) for some δ > 0
• Local minimum


  A function f attains a local minimum at x0 if f(x) ≥ f(x0) for all x in (f(x0) - δ, f(x0) + δ) for some δ > 0
• If a sequence {xn} is convergent, then it is bounded.
• and by definition of M, |%S| − |%| ≤ |%S − %| < 1. This implies that |%S| < 1 + |%|
• Define ε = max{1 + |%|, |%R|, |%\|, … , |%|}

  Then ∀ n ∈ ℕ, |%S| ≤ ε