4.2: The Mean Value Theorem:
Many results rely on this one fact.
Where a function f satisfies these 4 rules:
1. F is continuous at the closed interval [a,b].
2. F is differentiable on the open interval (a,b).
3. f(a) = f(b) there is a number c, so that f'(c) = 0. (tangent is horizontal)
There are 3 cases to this function for these 3 facts!
1. f(x) = k, f'(x) = 0 The function is a Refer to (a)
2. f(x) > f(a) in Extreme Value The function is Refer to (b)
(a,b) Theorem: There is a max never below f(a) and (c - both
in [a,b] 2 & 3)
f'(c) = 0
3. f(x) < f(a) in Extreme Value The function is Refer to (b)
(a,b) Theorem: There is a min never above f(a) and (c - both
at [a,b] 2 & 3)
f'(c) = 0
We use Rolle's Theorem in proving the Mean Value Theorem. Ex. Apply Rolle's Theorem to the position function: s = f(t) of a moving object.
If the object is in the same place at 2 different times, then f(a) = f(b)
There is an instance between a and b where velocity is 0. f'(c) = 0.
o Ex. Think of a ball thrown upwards.
Ex. Prove that the equation x + x - 1 = 0 has only one real root.
f(x) = x + x - 1
Let a = 0, and b = 1, to create interval [a,b] - We want the smallest possible difference from 0.
f(0) = -1 which is < 0
f(1) = 1 which is > 0
Therefore, there is a number c between a and b such that f'(c) = 0. There is a root.
But! Since f'(x) = 3x + 1 >= 1 for all x, since (x >= 0), f'(x) can never be 0.
Therefore, this equation cannot have 2 real roots.