# important questions class 12 maths chapter 6

## Important Questions for CBSE Class 12 Maths Chapter 6 – Application of Derivatives

Important Questions for Class 12 Maths Chapter 6 – Applications of Derivatives are provided here. The questions are based on the updated CBSE board syllabus. These important questions help students  achieve high marks in the Class 12 board examination. The important questions provided here are worth 1, 2, 4, and 6 marks.

Class 12 Maths Chapter 6 – Application of Derivative, covers important mathematical concepts such as tangents and normals, rate of change, maxima and minima, increasing and decreasing functions, and some simple problems that illustrate the fundamental concept of derivative and its application in real-life situations.

You can also find CBSE Class 12 Maths Important Questions Chapter-by-Chapter Important Questions here:

 CBSE Class 12 Maths Important Questions Sr No Chapter No Chapter Name 1 Chapter 1 Relations and Functions 2 Chapter 2 Inverse Trigonometric Functions 3 Chapter 3 Matrices 4 Chapter 4 Determinants 5 Chapter 5 Continuity and Differentiability 6 Chapter 6 Application of Derivatives 7 Chapter 7 Integrals 8 Chapter 8 Application of Integrals 9 Chapter 9 Differential Equations 10 Chapter 10 Vector Algebra 11 Chapter 11 Three Dimensional Geometry 12 Chapter 12 Linear Programming 13 Chapter 13 Probability

## CBSE Class 12 Maths Chapter-6 Important Questions

Some of the key questions from Chapter 6 – Application of Derivative Class 12 Maths are provided below, along with step-by-step solutions. Students can improve their final examination scores by practising the problems listed below, as they are important in the examination.

• If a quantity y varies with another quantity x in accordance with some rule y fx = (), then dy/dx (or f x ′()) represents the rate of change of y with respect to x.
• dy/dx x=x0 (or 0 f x ′()) denotes the rate of change of y in relation to x at x = x0
• If two variables x and y vary in relation to another variable t, i.e., if x=f(t) and y= g(t) then the Chain Rule applies dy/dx = dy/dt / dx/dt =, if dx/dt ≠ 0.
• A function f is said to-
• increase on an interval (a, b),
• x1 < x2 in (a, b) ⇒ f(x1) < f(x2) for all x1, x2 ∈ (a, b).
• Alternatively, if f ′(x) = 0 for each x in the equation (a, b)
• decrease on (a,b) if x1<x2 in (a, b) ⇒ f(x1) > f(x2) for all x1, x2 ∈ (a, b) (a, b).
• remain constant in (a, b), if f (x) = c for all x (a, b).
• The tangent equation at (x0,y0) to the y = f (x) curve is given by y-y0 = dy/dx](x0, y0) x-x0
• If dy/dx does not exist at 0 0 (,) x y, the tangent at this point is parallel to the y-axis, with the equation x = x0

The First Derivative Test: Assume f is a function defined on the open interval “I”. Let f be continuous in “I” at a critical point c. Then,

• If the sign of f ′(x) shifts from positive to negative as x increases through c, In other words, if f ′(x) > 0 at all points sufficiently close to and to the left of c, and f ′(x) 0 at all points sufficiently near and to the right of c, then c is a local maxima point.
• If f ′(x) changes sign from negative to positive as x increases through c, that is, if f ′(x) 0 at all points sufficiently close to and to the left of c and f ′(x) > 0 at all points sufficiently close to and to the right of c, then c is a point of local minima.
• If f ′(x) does not change the sign as x increases through c, then c is not a point of local maxima or minima. In fact, such a point is referred to as a point of inflection.

Second Derivative Test: Allow f to be a function defined on the interval I and c I. At c, let f be twice differentiable, then

•  x = c is a point of local maxima If f ′(c) = 0 and f ′′(c) = 0. The value f (c) is the maximum local value of f.
• If f ′(c) = 0 and f ′′(c) > 0, x = c is a point of local minima. f (c) is the local minimum value of f in this case.
• If f ′(c) and f ′′(c) are both 0, the test fails. In this case, we return to the first derivative test to determine whether c is a point of maxima, minima, or inflexion.

Q1.Find the intervals in which the function f given by f(x) = 4x3 – 6x2 – 72x + 30 is (a) strictly increasing (b) strictly decreasing.

Opt.

We have f(x) = 4x3 – 6x2 – 72x + 30  or f'(x) = 12x2 – 12x – 72

= 12(x2 – x -6) = 12(x – 3) (x + 2).

Therefore, f'(x) = 0 gives x = – 2 , 3.

There are three disjoint intervals, ( 2), (? 2, 3) and (3).

f'(x) > 0 for all x ? (2) and (3),

f'(x) < 0 for all x  ( 2, 3)

Ans.

We have f(x) = 4x3 – 6x2 – 72x + 30 or f'(x) = 12x2 – 12x – 72

= 12(x2 – x -6) = 12(x – 3) (x + 2).

Therefore, f'(x) = 0 gives x = – 2 , 3.

There are three disjoint intervals, (2), ( 2, 3) and (3).

(x) > 0 for all x  ( 2) and (3),

(x) < 0 for all x ( 2, 3)

Q2.

Opt.

## CBSE Class 12 Maths Important Questions

### 1. What are Maxima and Minima?

Minima is the lowest points on a graph while, Maxima is the highest points in a graph.

A derivative function can be used to calculate the lowest and highest points of a curve in a graph or to find the turning point.

When x = a and f (x) f (a), f (x) has an absolute maximum value for every x in the domain. Furthermore, point an is the maximum value of f.

When x = a and f (x) f (a), then f (x) has a relative minimum value for each x in some open interval (p, q).

When x = a and f (x) f (a), f (x) has an absolute minimum value for each x in the domain. Furthermore, point a represents the minimum value of f.

When x = a and f (x) f (a. for each x in some open interval (p, q), f (x) has a relative maximum value.

### 2. What is the Rate of Change of a Quantity?

The most important and widespread application of derivatives is the rate of change of a quantity. For example, if one wants to calculate the rate of change of a cube’s volume on the decreasing side, one can use the derivative form dy / dx.

dy in this equation represents the rate of change of a cube’s volume. dx, on the other hand, represents the change in the cube’s sides.

### 3. What are Applications of Derivatives in Mathematics?

The derivatives of functions like implicit functions, trigonometric functions, and logarithmic functions have numerous applications. These applications can be found in mathematical concepts as well as real-life scenarios. Among these applications are:

• Decreasing and increasing functions
• Newton’s method
• Linear approximation
• Rate of change of a quantity
• Maximum and minimum values
• Normal and tangent to a curve