Class 12 Math Chapter 6 Application of Derivatives Notes (Handwritten Notes PDF)

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Application of Derivatives Handwritten Notes PDF

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Key Points

Application of Derivatives If a quantity y varies with another quantity x, satisfying some rule (y=f(x)), then dx/dy (or f'(x)) represents the rate of change of y with respect to x at x=x₀. OR, f'(x₀) represent the rate change of y with respect to x at x=x₀

Maxima and Minima

Let f be a function defined on an interval I. Then

(i) f have maximum value in I if there exists a point c in I such that, f(c)>f(x) ∀ x ∈ I. The number f(c) is called max. value of f in I. Point c is called a point of max. value of f in I.

(ii) f have minimum value in I if there exists a point c in I such that, f(c)<f(x) ∀ x ∈ I. The number f(c) is called max. value of f in I. Point c is called a point of min. value of f in I.

(iii) f have an extreme value in I, if there exists a point c in I such that f(c) is either a maximum value or a minimum value of f in I. The number f(c) is called an extreme value of f in I. The point c is called an extreme point.

Local Maxima and local minima Let f be a real valued function and let c be an interior point in the domain of f. Then,

(a) c is called a point of local maxima if there is an h>0 such that, f(c)>f(x) ∀ x ∈ (c-h, c+h). The value of f(c) is called local max. value of f.

(b) c is called a point of local minima if there is an h>0 such that, f(c)<f(x) ∀ x ∈ (c-h, c+h). The value of f(c) is called local min. value of f.

Turning points: The points at which a function changes its nature from decreasing to increasing or vice versa are called turning points.

Critical points: A point c in the domain of a function f at which either f'(c)=0 or f is not differentiable, is called critical point of f.

First derivative test: Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then,

• If f'(x) changes sign from +ve to -ve as x increases through c, then c is a point of local maxima.
• If f'(x) changes sign from -ve to +ve as x increases through c, then c is a point of local minima.
• If f'(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. Such a point is called point of inflection.

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Summary