Easy Concept Development with RD Sharma Class 12 Maths Solutions Chapter 15 - Mean Value Theorems

This chapter primarily focuses on the application and verification of Mean Value Theorems. The key topics you will master using the RD Sharma solutions are:

• Rolle’s Theorem: Understanding its three conditions (continuity, differentiability, and f(a) = f(b)) and finding a point 'c' where f'(c) = 0.
• Lagrange’s Mean Value Theorem (LMVT): Applying its conditions (continuity and differentiability) to find a point 'c' such that f'(c) = (f(b) - f(a)) / (b - a).
• Geometrical Interpretation: Visualising what Rolle's Theorem and LMVT represent in terms of tangents and secant lines on a curve.
• Problem-Solving: Solving a wide variety of questions to verify the theorems for different types of functions (polynomial, trigonometric, etc.).

To verify Rolle's Theorem for a function f(x) on a closed interval [a, b], you must follow these sequential steps as shown in the RD Sharma solutions:

1. Check for Continuity: Confirm that the function f(x) is continuous on the closed interval [a, b]. For polynomial functions, this is always true.
2. Check for Differentiability: Confirm that the function f(x) is differentiable on the open interval (a, b).
3. Check Endpoint Values: Verify that f(a) = f(b).
4. Find the Derivative: If all three conditions are met, find the derivative, f'(x).
5. Solve for 'c': Set f'(c) = 0 and solve for c. The value of c must lie within the open interval (a, b) for the theorem to be verified.

The main difference lies in the conditions and the conclusion. Rolle's Theorem is a special case of LMVT.

• Rolle's Theorem requires three conditions: the function must be continuous, differentiable, and have equal values at the endpoints (f(a) = f(b)). It guarantees a point 'c' where the tangent to the curve is horizontal (f'(c) = 0).
• Lagrange's MVT only requires two conditions: continuity and differentiability. It does not require f(a) = f(b). It guarantees a point 'c' where the tangent to the curve is parallel to the secant line connecting the endpoints (a, f(a)) and (b, f(b)).

Checking for continuity and differentiability is the most crucial first step because these are the foundational requirements of both theorems. If a function is not continuous on [a, b], it has a break, and if it's not differentiable on (a, b), it has a sharp corner. In either scenario, the geometric guarantee of the theorems (the existence of a specific tangent line) is not valid. Skipping this check can lead you to incorrectly apply the theorem and arrive at a wrong conclusion, which is a common pitfall in exams.

The geometric interpretation provides a visual meaning to the theorem. It states that for any smooth, continuous curve between two points, there exists at least one point on the curve where the instantaneous rate of change (slope of the tangent line) is exactly equal to the average rate of change over the entire interval (slope of the secant line connecting the endpoints). In simpler terms, there is always a point 'c' where the tangent line is perfectly parallel to the chord joining the start and end points of the curve.

It depends on the interval. For example, the function f(x) = |x| is not differentiable at x=0 because it has a sharp corner. Therefore, you cannot apply Mean Value Theorems on any interval that includes 0, such as [-1, 1], because the differentiability condition would fail. However, on an interval where the function is smooth, like [1, 3], the theorems can be applied because f(x) = x in that domain, which is both continuous and differentiable.