Mean Value Theorems (Rolle's)

1. Statement of Rolle's Theorem

If a function f satisfies the three conditions

f is continuous on the closed interval [a, b],
f is differentiable on the open interval (a, b), and
f(a) = f(b) (the values at the endpoints are equal),

then there exists at least one point c in the open interval (a, b) such that

f'(c) = 0, a < c < b

In words: between two points of equal height on a smooth curve, the tangent must be horizontal somewhere in between. Rolle's theorem is the special case of Lagrange's Mean Value Theorem in which the chord is horizontal; it is the geometric heart of all the mean value theorems.

2. Geometric and Algebraic Interpretations

Geometric: if the graph starts and ends at the same level, then somewhere the curve must "turn around" — there is a horizontal tangent (a maximum, a minimum, or a stationary point of inflection).
Algebraic: between two roots of a differentiable function there lies at least one root of its derivative. In general, between two roots of f there is a root of f′, between two roots of f′ a root of f″, and so on — a fact used to locate roots.
Physical: if a particle returns to its starting height, there was an instant when its vertical velocity was zero.

   Verify Rolle's theorem for  f(x) = x² - 4  on  [-2, 2]

   (i)   f is a polynomial ⇒ continuous on [-2, 2] and differentiable on (-2, 2) ✓
   (ii)  f(-2) = 4 - 4 = 0  and  f(2) = 4 - 4 = 0   ⇒ f(a) = f(b) ✓
   (iii) f′(x) = 2x = 0   ⇒   c = 0 ,   and  -2 < 0 < 2 ✓
   ⇒  Rolle's theorem is verified, with c = 0 (here f″(0) = 2 > 0, so it is the minimum).

6. Why Every Hypothesis Is Needed — Counter-Examples

Failure of hypothesis	Example	What goes wrong
Not continuous	f(x) = 1/x on [−1, 1]	f is not defined (hence not continuous) at x = 0 — there is a vertical asymptote and no admissible c
Not differentiable inside	f(x) = mod x on [−1, 1]	f(−1) = f(1) = 1, but the corner at x = 0 means f′ does not exist there — no c with f′(c) = 0
Endpoint values unequal	f(x) = x on [0, 1]	f(0) = 0 but f(1) = 1; f′(x) = 1 never vanishes

(The middle example is the modulus function, whose graph has a sharp corner at the origin — writing it as |x| inside a table would break the table, hence "mod x" above.) Each of these is a favourite "why does Rolle fail?" question, and they also show why Lagrange's theorem (next topic) needs only the first two hypotheses.

7. Consequences and Uses

Between two zeros of f there is a zero of f′ — used to prove that an equation has at most one root in an interval, or to locate roots numerically.
Multiplicity of roots: if f has n distinct real roots, f′ has at least n − 1 real roots (a repeated root of f is also a root of f′).
It is the special case of Lagrange's Mean Value Theorem with f(a) = f(b), and is used in Lagrange's proof of Taylor's theorem.
Constant function test: if f′(x) = 0 for every x in an interval, then f is constant there (proved by Rolle/Lagrange).
Used to prove inequalities by bounding f′ over an interval.

8. Exam Points

State Rolle's theorem with all three conditions — 3 marks, almost certain.
Verify Rolle's theorem for a given function and find the value(s) of c — 5-7 marks, the most frequent question of this topic.
Give the geometric interpretation — 3 marks.
Show by counter-example that each hypothesis is necessary (1/x, mod x, and f(x) = x) — 5 marks, frequent.
Use Rolle's theorem to prove that between two roots of f there is a root of f′ — 5 marks.
Prove that f′(x) = 0 everywhere on an interval implies f is constant there — 3-5 marks.

Mean Value Theorems (Rolle's)