Numbers and Functions Steps into Analysis

Concurrent reading: Burkill ch. 4 and §5.8, Courant and John ch. 5.

Further reading: Mason, Spivak chs 11, 20.

Both Rolle's Theorem and the Mean Value Theorem are geometrically transparent. Each claims, with slightly more generality in the case of the Mean Value Theorem, that for the graph of a differentiable function, there is always a tangent parallel to a chord. It is something of a surprise to find that such intuitive results can lead to such powerful conclusions: namely, de l'Hôpital's rule and the existence of power series convergent to a wide family of functions.

Rolle's Theorem

1 Sketch the graphs of some differentiable functions

f : [a, b] → ℝ,

for which f (a) = f (b).

Can you find a point c, with a < c < b, such that f′ (c) = 0, for each function f which you have sketched?

What words could you use to describe the point c in relation to the function f?

2 Must any differentiable function

f: [a, b] → ℝ

have a maximum and a minimum value? Why?

Must any such function, for which f (a) = f (b), have a maximum and a minimum value in the open interval (a, b)?

3 If f: [a, b] → ℝ is differentiable and, for some c such that a < c < b, f(x) ≤ f(c) for all x ∈ [a, b], must f′(c) = 0? Why?

4 If f: [a, b] → ℝ is differentiable,

f(a) = f(b),

M = sup{f(x)| a ≤ x ≤ b},

and m = inf{f(x)| a ≤ x ≤ b},

use qn 2 to show that there is a c with a < c < b for which f(c) = M, or a c for which f(c) = m, or possibly distinct cs satisfying each of these conditions.