Lagrange's

Lagrange's (Mean Value Theorem)

1. Statement

If a function f satisfies

1. f is continuous on the closed interval [a, b], and
2. f is differentiable on the open interval (a, b),

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

Notice that the third hypothesis of Rolle's theorem (f(a) = f(b)) is not needed — this is the main reason Lagrange's theorem is the workhorse of the chapter.

2. Geometric Meaning

The quantity [f(b) − f(a)]/(b − a) is the slope of the chord joining the endpoints (a, f(a)) and (b, f(b)). The theorem says: on a smooth curve, somewhere between the endpoints the tangent is parallel to the chord. Rolle's theorem is the special case where the chord is horizontal.

3. Proof (by constructing a function for Rolle)

Define the auxiliary function

This g is continuous on [a, b] and differentiable on (a, b) (because f is, and the rest is a linear function), and

• g(a) = f(a) − f(a) − 0 = 0,
• g(b) = f(b) − f(a) − [f(b) − f(a)] = 0.

So g(a) = g(b) and Rolle's theorem applies: there is c in (a, b) with g′(c) = 0. But

which is exactly Lagrange's statement.

4. Equivalent Forms Worth Writing

• f(b) = f(a) + (b − a) f′(c) — "the increment equals derivative × interval length at some interior point".
• With b = a + h and c = a + θh (0 < θ < 1): f(a + h) = f(a) + h f′(a + θh) — the form used throughout numerical analysis.
• Differential approximation: Δy ≈ f′(x) Δx for small Δx.

   Verify Lagrange's MVT for  f(x) = x²  on  [1, 3]

   (i)   polynomial ⇒ continuous on [1,3], differentiable on (1,3) ✓
   (ii)  Chord slope = [f(3) - f(1)]/(3 - 1) = (9 - 1)/2 = 4
   (iii) f′(x) = 2x  ⇒  2c = 4  ⇒  c = 2  and  1 < 2 < 3 ✓
   ⇒  verified, with c = 2 (the midpoint here, because a parabola has constant second derivative).

8. Consequences of Lagrange's Theorem

1. Monotonicity test: f′(x) > 0 for all x in an interval ⇒ f is strictly increasing there; f′(x) < 0 ⇒ strictly decreasing; f′(x) = 0 ⇒ f is constant on that interval.
2. Root-location tool: if f′ has no zero in (a, b), then f can have at most one zero there; in general between two zeros of f there is a zero of f′.
3. Error and Lipschitz bounds: if |f′(x)| ≤ M on [a, b], then |f(b) − f(a)| ≤ M(b − a) — used to bound numerical errors.
4. It gives the linear approximation f(x + h) ≈ f(x) + h f′(x), and leads to Taylor's theorem with the Lagrange remainder.
5. It is used (with Cauchy's MVT) to prove L'Hôpital's rule.

9. Exam Points

• State Lagrange's Mean Value Theorem and give its geometric interpretation — 3-5 marks, almost certain.
• Prove Lagrange's MVT using Rolle's theorem — 5 marks, very frequent.
• Verify the theorem for a given function and find c — 5-7 marks, the most frequent numerical question of this topic.
• Use the MVT to prove an inequality such as (b − a)/b < log(b/a) < (b − a)/a — 5-7 marks, frequent.
• Prove that f′ = 0 on an interval implies f is constant; prove monotonicity from the sign of f′ — 5 marks.
• Write Lagrange's theorem as f(a + h) = f(a) + h f′(a + θh) and explain each symbol — 3 marks.
• Show that Rolle's theorem is a special case of Lagrange's theorem — 3 marks.