RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.3) Exercise 18.3 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.3) Exercise 18.3
1. What is the step-by-step method to solve questions on local maxima and minima in RD Sharma Class 12, Exercise 18.3?
The solutions for Exercise 18.3 primarily use the Second Derivative Test. The standard procedure is as follows:
- First, find the derivative of the given function, f'(x).
- Next, find the critical points by setting f'(x) = 0 and solving for x.
- Then, calculate the second derivative, f''(x).
- Substitute each critical point 'c' into the second derivative. If f''(c) < 0, the point is a local maximum. If f''(c) > 0, it's a local minimum.
If f''(c) = 0, the test fails, and you must use the First Derivative Test.
2. How do you find the critical points of a function when solving problems from Chapter 18?
To find the critical points of a function y = f(x), you must first calculate its derivative, f'(x). A critical point is a point 'c' in the domain of the function where either the derivative is zero (f'(c) = 0) or the derivative is not defined. For most polynomial and trigonometric functions in this chapter, you will find critical points by setting f'(x) = 0 and solving the resulting equation for x.
3. Are the methods used in Vedantu's RD Sharma Solutions for Chapter 18 aligned with the CBSE 2025-26 syllabus?
Yes, the solutions for RD Sharma Class 12 Chapter 18 are fully aligned with the CBSE 2025-26 syllabus. The methods, particularly the First and Second Derivative Tests for finding maxima and minima, are explained as per the curriculum guidelines for the "Applications of Derivatives" unit. This ensures you are preparing with methods that are acceptable in board exams.
4. What is a common mistake to avoid when using the Second Derivative Test for problems in Ex 18.3?
A very common mistake is concluding that a point is a point of inflection if the Second Derivative Test fails (i.e., if f''(c) = 0). This is incorrect. If f''(c) = 0, the test is simply inconclusive. You must then revert to the First Derivative Test to check the sign of f'(x) just before and after the critical point 'c' to determine if it is a local maximum, minimum, or a point of inflection.
5. In what situation is the First Derivative Test more useful than the Second Derivative Test when solving for local extrema?
The First Derivative Test is more useful in two main situations:
- When the second derivative is very complex or tedious to calculate. In such cases, checking the sign of f'(x) around the critical point is often quicker.
- When the Second Derivative Test fails, meaning f''(c) = 0 at a critical point 'c'. The First Derivative Test is the necessary next step to classify the point.
It is also the only method applicable for critical points where the derivative does not exist.
6. How do you identify a point of inflection, and how does it differ from a local minimum or maximum?
A point of inflection is a point on a curve where the concavity changes (from concave up to concave down, or vice versa). While a local minimum/maximum occurs where f'(x) = 0 and f''(x) has a definitive sign, a point of inflection 'c' typically occurs where f''(c) = 0 and the sign of f''(x) is different on either side of 'c'. Unlike a local extremum, which is a 'peak' or 'trough', a point of inflection is where the curve 'flattens' momentarily before continuing its rise or fall.
7. Why is it important to solve every problem in RD Sharma's Exercise 18.3?
Solving every problem in Exercise 18.3 is crucial for mastering the application of the Second Derivative Test. The exercise provides a wide variety of functions—polynomial, trigonometric, and logarithmic—which helps in building proficiency and speed. This practice ensures you can accurately find critical points and correctly interpret the results of the derivative tests, a key skill for the CBSE board exams.
