LECTURE 3 ROLLE'S AND LAGRANGE'S MEAN VALUE THEOREM CHAPTER 8 CLASS 12

Introduction to Rolle's and Lagrange's Mean Value Theorems
In calculus, Rolle's Theorem and Lagrange's Mean Value Theorem (LMVT)** are fundamental results that provide important insights into the behavior of differentiable functions. These theorems are essential tools in analyzing and understanding the behavior of functions, and they lay the groundwork for more advanced topics in differential calculus.
1. Rolle's Mean Value Theorem (RMVT): - Rolle's Mean Value Theorem is a special case of the Mean Value Theorem. It states that if a function satisfies certain conditions, then there is at least one point where the derivative (slope) of the function is zero.
Conditions for Rolle's Theorem: (i) The function f(x) is continuous on the closed interval ([a, b].
(ii) The function f(x) is differentiable on the open interval (a, b).
(iii) The function values at the endpoints are equal, i.e., f(a) = f(b).
Conclusion of Rolle's Theorem: - There exists at least one point c in the interval (a, b) such that f’(c) = 0
This means that the function has a horizontal tangent (slope = 0) at some point c within the interval.
Example: -
For the function f(x) = x^2 - 4x + 4 on the interval [0, 4]
• f(0) = f(4) = 4, so the endpoints are equal.
• The function is continuous and differentiable on the given interval.
• By Rolle's Theorem, there exists some c in (0, 4) such that f'(c) = 0.
2. Lagrange's Mean Value Theorem (LMVT): - Lagrange's Mean Value Theorem generalizes Rolle's Theorem by relaxing the condition that the function values at the endpoints must be equal.
Conditions for LMVT: - (i) The function f(x) is continuous on the closed interval [a, b].
The function f(x) is differentiable on the open interval (a, b).
Conclusion of LMVT: - There exists at least one point c in the interval (a, b) such that f'(c) = f(b) - f(a)/b - a.
This equation means that the slope of the tangent at some point c is equal to the average slope of the function over the interval [a, b].
Example: -
For the function f(x) = x^2 on the interval [1, 3]
• The function is continuous and differentiable on the given interval.
• By LMVT, there exists some c in (1, 3) such that f'(c) = f(3) - f(1)/3 - 1 = 9 - 1/2 = 4
• The derivative f'(x) = 2x . So, 2c = 4 and c = 2.
These theorems are crucial in understanding the behavior of functions, proving the existence of roots, and in various applications across mathematics and physics.
3. Note: -
(i) Constant Functions [ eg. f(x) = k ] is continuous and differentiable everywhere.
(ii) Polynomial Functions [ eg. f(x) = ax^2+bx+c ] is continuous and differentiable everywhere.
(iii) Exponential Functions [ eg. f(x) = a^x or e^x ] is continuous and differentiable everywhere.
(iv) Sine Functions [ eg. f(x) = Sinx ] is continuous and differentiable everywhere.
(v) Cosine Functions [ eg. f(x) = Cosx ] is continuous and differentiable everywhere.
(vi) Tangent, Cotangent, Secant and Cosecant are continuous and differentiable in their domains.
(vii) Inverse Trigonometric Functions are continuous and differentiable in their domains.
(viii) Logarithmic Functions are continuous and differentiable in their domains.
(ix) Sum, difference, product and quotient of continuous functions are continuous in the given intervals.
(x) Sum, difference, product and quotient of differentiable functions are differentiable in the given intervals.
(xi) Every differentiable function is continuous. i.e. if a function is not continuous at a point then the function is not differentiable at that point.

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