NCERT Solutions for Application of Derivatives Class 12 Notes Maths Chapter 6
This chapter Application of derivatives mainly features a set of topics just like the rate of change of quantities, Increasing and decreasing functions, Tangents and normals, Approximations, Maxima and minima, and lots more. We are going to discuss the important concepts of the chapter application of derivatives.
1. A function f is said to be
Increasing only if on an interval of (a, b) if x1 < x2
Decreasing on (a,b) if x1 < x2
Let the constant (a, b), if f (x) equals c for all x ∈ (a, b), where c is the constant.
2. First Derivative Test.
3. Second Derivative Test.
The main topics which are covered for the NCERT Solutions for Application of Derivatives Class 12 notes, Chapter 6 are:
NCERT Solutions for Application of Derivatives Class 12 Notes Mathematics Chapter 6
Maximum and Minimum Value
Let f be the function which is defined on an interval I. So,
f is claimed to possess a maximum value in I, if there exists some extent c in I such: f(c) > f(x), ∀ x ∈ I. The number f(c) is named the utmost value of f in I and therefore the point c is named to some extent a maximum value of f in I.
f is said to be having a minimum value in I, and if there exists a point c in I so f(c) < f(x), ∀ x ∈ I. The number f(c) is named the minimum value of f in I and therefore the point c is named to some extent the minimum value of f in I.
f is claimed to possess an extreme value in I, if there exists some extent c in I such f(c) is either a maximum value or a minimum value of f in I. The number f(c) is named an extreme value off in I and therefore the point c is named an extreme.
Important Points of Applications of Derivatives
Through the graphs, we will even find the maximum/minimum value of a function to some extent at which it's not even differentiable.
Every monotonic function makes sure that its maximum/minimum value is at the endpoints of the domain of the definition of the function.
Every continuous function on a bounded interval features a maximum and a minimum value.
Let f be a function which is defined on an unbounded interval which is I. Suppose cel is any point. If f has local maxima or local minima at x = c, then either f'(c) = 0 or f isn't differentiable at c.
Critical Point: to some extent c within the domain of a function f at which either f'(c) = 0 or f isn't differentiable, is named a juncture of f.
First Derivative Test: Let f be a function defined on an unbounded interval which is I and f be the continuous of a juncture c in I. So, if f'(x) changes sign from positive to negative as x increases through c, then c may be a point of local maxima.
if f'(x) changes sign from negative to positive as x increases through c, then c may be a point of local minima.
if f'(x) doesn't change sign as x increases through c, then c is neither some extent of local maxima nor some extent of local minima. Such some extent is named some extent of inflection.
Second Derivative Test: Let f(x) be a function that is defined on an interval known as I and c ∈ I. Let f be two times differentiable at c. So,
x = c is a point for the local maxima, if f'(c) equals 0 and f''(c) < 0.
x = c is a point for the local minima, if f'(c) equals 0 and f''(c) > 0.
The test will fail if f'(c) equals 0 and f''(c) equals 0.
FAQs on Application of Derivatives For Class 12
1. What are the main concepts covered under Application of Derivatives for Class 12 as per the 2025–26 CBSE syllabus?
The key concepts in the Application of Derivatives chapter for Class 12 include rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and maxima and minima. Students should also focus on derivative tests and graphical interpretations required for board exams.
2. How do you determine if a function is increasing or decreasing using derivatives?
To check if a function is increasing on an interval, calculate the derivative f'(x). If f'(x) > 0 throughout the interval, the function is increasing. For a decreasing function, f'(x) < 0 on the interval. Use this rule to solve CBSE exam questions on monotonicity.
3. What is the First Derivative Test and how is it applied in board questions?
The First Derivative Test involves finding points where f'(x) = 0 (critical points). Analyze the sign of f'(x) around these points: if it changes from positive to negative, there is a local maximum; if it changes from negative to positive, there is a local minimum. This method is essential for solving board-type maxima and minima problems.
4. How does the Second Derivative Test help in finding maxima and minima?
At a critical point where f'(c) = 0, check the value of the second derivative f''(c). If f''(c) > 0, there is a local minimum; if f''(c) < 0, there is a local maximum. This test is quicker if it applies, but may fail if f''(c) = 0.
5. Why is the concept of turning points important for board marking schemes?
Turning points indicate where the function changes from increasing to decreasing (or vice versa), which often correspond to maxima or minima. Identification and analysis of turning points are frequently asked in Class 12 board exams and carry significant weightage in 4-mark or 5-mark questions.
6. What are the common mistakes students make when applying the First and Second Derivative Tests?
Students often forget to verify if a critical point is within the domain, or ignore endpoints for closed intervals. Another mistake is assuming the Second Derivative Test always works; if f''(c) = 0, the test is inconclusive and further analysis is needed through the First Derivative Test or graph inspection.
7. How are tangents and normals to curves found using derivatives in CBSE Class 12 exams?
The slope of the tangent at a point is given by f'(x). The equation of the tangent can be written as y - y₁ = f'(x₁)(x - x₁). The normal to the curve has a slope of -1 / f'(x₁). Board exam questions may ask for tangent or normal equations at particular points on the curve.
8. In real-world applications, how do derivatives help in finding rates of change?
Derivatives are used to compute rates such as velocity (rate of change of distance with respect to time) or growth rates in economics and biology. These applications are part of the Class 12 syllabus and sometimes appear as HOTS (Higher Order Thinking Skills) questions in CBSE exams.
9. Are absolute maxima/minima and local maxima/minima evaluated differently in board exam questions?
Yes. Absolute (global) maxima/minima refer to the highest or lowest value on an interval, while local maxima/minima occur at a point compared to immediate neighbors. For closed intervals, check critical points and endpoints for absolute values; for local values, use derivative tests near the point. Board marks may be allotted for both interpretations.
10. How should students present stepwise solutions for Application of Derivatives questions to maximize marks in the CBSE 2025–26 exam?
- Clearly state what is to be found (e.g., maxima/minima, tangent equation).
- Find the derivative and equate to zero to locate critical points.
- Apply the First or Second Derivative Test to classify the points.
- Evaluate the function at critical points and endpoints (if required).
- Write all working steps and final answers neatly, as CBSE marking schemes reward logical, stepwise presentation.
