What is the Proof of Lagrange's Mean Value Theorem?

S7-SA1-0592

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The proof of Lagrange's Mean Value Theorem (LMVT) relies on Rolle's Theorem. It involves constructing a special auxiliary function that satisfies the conditions of Rolle's Theorem, which then helps us show that there's a point where the tangent to the curve is parallel to the secant line connecting the endpoints.

Simple Example

Quick Example

Imagine you're driving your scooter from your home to a friend's house. You cover 10 km in 20 minutes. LMVT says that at some point during your trip, your instantaneous speed (what your speedometer shows) must have been exactly equal to your average speed (10 km / 20 min = 0.5 km/min). The proof helps us mathematically confirm this 'some point' exists.

Worked Example

Step-by-Step

Let's prove LMVT for a function f(x) on the interval [a, b].

1. **Understand LMVT:** LMVT states that if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one point 'c' in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

---2. **Construct an Auxiliary Function:** We define a new function, let's call it g(x):
g(x) = f(x) - Kx
where K is a constant. Our goal is to make g(x) satisfy Rolle's Theorem.

---3. **Apply Rolle's Theorem Condition:** For Rolle's Theorem, we need g(a) = g(b).
g(a) = f(a) - Ka
g(b) = f(b) - Kb

---4. **Solve for K:** Set g(a) = g(b):
f(a) - Ka = f(b) - Kb
Kb - Ka = f(b) - f(a)
K(b - a) = f(b) - f(a)
K = (f(b) - f(a)) / (b - a)

---5. **Define g(x) with K:** Now, substitute the value of K back into g(x):
g(x) = f(x) - [(f(b) - f(a)) / (b - a)]x

---6. **Check Rolle's Conditions for g(x):**
* g(x) is continuous on [a, b] because f(x) and Kx are continuous.
* g(x) is differentiable on (a, b) because f(x) and Kx are differentiable.
* We specifically chose K so that g(a) = g(b).

---7. **Apply Rolle's Theorem:** Since g(x) satisfies all conditions of Rolle's Theorem, there must exist at least one point 'c' in (a, b) such that g'(c) = 0.

---8. **Differentiate g(x):**
g'(x) = f'(x) - K
Set g'(c) = 0:
f'(c) - K = 0
f'(c) = K

---9. **Substitute K back:**
f'(c) = (f(b) - f(a)) / (b - a)

This proves Lagrange's Mean Value Theorem. We found a point 'c' where the derivative f'(c) equals the slope of the secant line.

Why It Matters

Understanding this proof is key for advanced mathematics, which forms the backbone of AI algorithms for optimization and machine learning models. Engineers use it to analyze rates of change in systems, helping design everything from efficient EV batteries to rocket trajectories. Even in finance, understanding average rates of change and instantaneous rates helps predict market trends.

Common Mistakes

MISTAKE: Assuming the function itself must be zero at endpoints like in Rolle's Theorem. | CORRECTION: In LMVT, the *values* of the function at endpoints (f(a) and f(b)) don't need to be equal. We construct an *auxiliary function* whose values *are* equal at the endpoints.

MISTAKE: Forgetting to check the continuity and differentiability conditions for the original function f(x) before applying LMVT. | CORRECTION: Always state and verify that f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) first. These are crucial prerequisites.

MISTAKE: Confusing the constant 'K' with 'c'. | CORRECTION: 'K' is a constant value representing the slope of the secant line, calculated from the endpoints. 'c' is the *specific point* within the interval where the instantaneous slope (derivative) equals 'K'.

Practice Questions

Try It Yourself

QUESTION: State the two main conditions a function must satisfy for Lagrange's Mean Value Theorem to be applicable on an interval [a, b]. | ANSWER: 1. The function must be continuous on the closed interval [a, b]. 2. The function must be differentiable on the open interval (a, b).

QUESTION: If f(x) = x^2 on the interval [1, 3], find the value of 'c' that satisfies LMVT. | ANSWER: f(x) = x^2, f'(x) = 2x. f(1) = 1, f(3) = 9. (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = 8 / 2 = 4. Set f'(c) = 4, so 2c = 4, which means c = 2.

QUESTION: The proof of LMVT uses another important theorem. Which theorem is it, and what condition does the auxiliary function satisfy to make that theorem applicable? | ANSWER: It uses Rolle's Theorem. The auxiliary function g(x) is constructed such that g(a) = g(b), which is a key condition for Rolle's Theorem.

MCQ

Quick Quiz

The auxiliary function g(x) = f(x) - Kx in the proof of LMVT is designed to satisfy which key condition of Rolle's Theorem?

g(x) is always positive

g'(x) = 0 for all x

g(a) = g(b)

g(x) has a maximum or minimum at 'c'

The Correct Answer Is:

The auxiliary function g(x) is specifically constructed so that its values at the endpoints, g(a) and g(b), are equal. This is the crucial condition that allows Rolle's Theorem to be applied to g(x).

Real World Connection

In the Real World

Imagine a drone delivering a package for a company like Flipkart or Amazon. If the drone travels a certain distance in a specific time, the proof of LMVT assures us that at some point, its instantaneous speed exactly matched its average speed for the entire trip. This concept helps engineers design flight paths and optimize delivery times, ensuring efficiency and safety.

Key Vocabulary

Key Terms

AUXILIARY FUNCTION: A specially created function used to help prove another theorem, often by applying a known theorem to it. | ROLLE'S THEOREM: A theorem stating that if a differentiable function has equal values at two points, it must have a stationary point (where the derivative is zero) somewhere between them. | SECANT LINE: A line connecting two points on a curve. Its slope represents the average rate of change. | TANGENT LINE: A line that touches a curve at a single point and has the same slope as the curve at that point (derivative).

What's Next

What to Learn Next

Next, you can explore applications of LMVT in solving inequalities and approximating function values. You can also look into Taylor's Theorem, which is an extension of LMVT and is super important in AI and numerical methods for approximating complex functions!