Maths Application of Derivatives Formulas: The CBSE Class 12 mathematics course is predominantly focused on calculus, and chapter 6 Application of Derivatives is one of the most important chapters. It’s the final chapter in book 1 and answers the lingering question many students have: what is the use of derivatives and the numerous formulas and identities?
Application of Derivatives is a long chapter and also important from an exam point of view as well. It consists of several theorems, rules and formulas that students have to memorise. Although it’s a difficult task, learning formulas helps simplify complex equations and solve problems quickly. You can check the CBSE Class 12 Maths Chapter 6 Application of Derivatives Formulas below.
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We have listed all the important formulas, definitions and properties of CBSE Class 12 Application of Derivatives here.
Increasing & Decreasing Function
A function ƒ is said to be
(a) increasing on an interval (a, b) if
x1 < x2 in (a, b) ⇒ ƒ (x1 ) ≤ ƒ(x2) ƒor all x1 , x2 ∈ (a, b).
Alternatively, if ƒ’(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if
x1 < x2 in (a, b) ⇒ ƒ (x1) ≥ ƒ (x2) for all x1 , x2 ∈ (a, b).
Alternatively, if ƒ ′(x) ≤ 0 for each x in (a, b)
Maxima & Minima
First Derivative Test
Let ƒ be a function defined on an open interval I and Let f be continuous at a critical point c in I. Then,
(i) If ƒ ′(x) changes sign from positive to negative as x increases through c, i.e., if ƒ ′(x) > 0 at every point sufficiently close to and to the left of c, and ƒ ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.
(ii) If ƒ ′(x) changes sign from negative to positive as x increases through c, i.e., if ƒ ′(x) < 0 at every point sufficiently close to and to the left of c, and ƒ ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.
(iii) If ƒ ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called the point of inflexion.
Second Derivative Test
Let ƒ be a function defined on an interval I and c ∈ I. Let ƒ be twice differentiable at c. Then,
(i) x = c is a point of local maxima if ƒ ′(c) = 0 and ƒ ″(c) < 0
The values ƒ (c) is local maximum value of ƒ .
(ii) x = c is a point of local minima if ƒ ′(c) = 0 and ƒ ″(c) > 0
In this case, ƒ (c) is local minimum value of ƒ .
(iii) The test fails if ƒ ′(c) = 0 and ƒ ″(c) = 0.
In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.
Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of ƒ in the interval, i.e., find points x where either ƒ ′(x) = 0 or ƒ is not differentiable.
Step 2: Take the end points of the interval.
Step 3: At all these points (listed in Step 1 and 2), calculate the values of ƒ .
Step 4: Identify the maximum and minimum values of ƒ out of the values calculated in Step 3. This maximum value will be the absolute maximum value of ƒ and the minimum value will be the absolute minimum value of f
Also Read
CBSE Class 12 Maths Syllabus 2023-24
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