RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.2) Exercise 15.2 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.2) Exercise 15.2
1. What is the primary focus of RD Sharma Class 12 Solutions for Chapter 15, Exercise 15.2?
This exercise primarily focuses on the application of Lagrange's Mean Value Theorem (L-MVT). The problems require you to verify the theorem for various functions over a given closed interval and then find the specific value of 'c' that the theorem guarantees exists within that interval.
2. What are the correct steps to verify Lagrange's Mean Value Theorem for a function f(x) on an interval [a, b]?
To correctly solve and verify L-MVT for a function f(x) on a closed interval [a, b] as per the CBSE curriculum, you must follow these sequential steps:
Step 1: Check for Continuity. First, you must state that the function f(x) is continuous on the closed interval [a, b]. For polynomial, sine, cosine, and exponential functions, this is always true.
Step 2: Check for Differentiability. Next, find the derivative f'(x) and state that the function is differentiable on the open interval (a, b).
Step 3: Apply the L-MVT Formula. If both conditions are met, the theorem states there is a point 'c' in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a).
Step 4: Solve for 'c'. Calculate the values of f(b) and f(a), substitute them into the formula, and solve the resulting equation for 'c'.
Step 5: Verify the Value of 'c'. Ensure that the value of 'c' you obtained lies strictly between 'a' and 'b'.
3. What is the geometric interpretation of Lagrange's Mean Value Theorem?
The geometric significance of L-MVT is a key concept for Class 12 Maths. It states that for a smooth, continuous curve between two endpoints, there is at least one point on the curve where the tangent line is parallel to the secant line connecting the endpoints. In simpler terms, the instantaneous rate of change (slope of the tangent) at some point 'c' is equal to the average rate of change over the entire interval (slope of the secant).
4. Why must a function be continuous and differentiable for L-MVT to apply?
These two conditions are the foundation of the theorem. Continuity on [a, b] ensures that the curve is unbroken, so a secant line connecting the endpoints can be drawn. Differentiability on (a, b) ensures the curve is smooth and has no sharp corners or vertical tangents. This guarantees that a unique, non-vertical tangent line can be drawn at every point inside the interval, making it possible to find one that is parallel to the secant.
5. What is the fundamental difference between Rolle's Theorem and Lagrange's Mean Value Theorem?
Lagrange's Mean Value Theorem is a more general form of Rolle's Theorem. The key difference is the third condition required for Rolle's Theorem: f(a) = f(b). This condition means the endpoints of the interval are at the same height. Consequently, Rolle's Theorem concludes that there must be a point 'c' where f'(c) = 0 (a horizontal tangent). L-MVT does not require f(a) = f(b) and concludes that the tangent at 'c' is parallel to the secant line, which is not necessarily horizontal.
6. What are the most common mistakes students make when solving L-MVT problems in Exercise 15.2?
The most frequent errors include:
Ignoring Conditions: Failing to explicitly state that the function is continuous and differentiable.
Calculation Errors: Simple mistakes while calculating f(a), f(b), or the derivative f'(x).
Verification Failure: The most critical mistake is finding a value for 'c' but forgetting to check if it lies within the open interval (a, b). If it doesn't, it is not a valid solution according to the theorem.
7. Can Lagrange's Mean Value Theorem be applied to a function like f(x) = |x| on the interval [-1, 1]?
No, L-MVT cannot be applied to f(x) = |x| on the interval [-1, 1]. While the function is continuous on [-1, 1], it is not differentiable at x = 0, which lies within the open interval (-1, 1). The graph has a sharp corner at x = 0, so a unique tangent cannot be drawn there. Since the differentiability condition fails, the theorem is not applicable.
