# Application of Derivatives Class 12 Notes Mathematics Chapter 6

### Chapter at a Glance

1. Increasing and Decreasing Functions:

2. To Find Absolute Maximum and Absolute Minimum in continuous function f (x) in [a, b]

(i) Evaluate f (x) at points in [a, b] where f ‘(x) = 0

(ii) Find f (a) and f (b)

Then, the maximum of these values is the absolute maximum and the minimum of these values is the absolute minimum of the function f (x)

3. To Find Local Maxima and Local Minima in continuous function f (x) in(a, b)

First Derivative Test

(i) Evaluate the values of x, where either f ‘(x) = 0.

Let x1, x2, x3, ……., xn are the values of x in [a, b], where f ‘(x) = 0.

(ii) Evaluate those values of x out of x1, x2, x3, ….., xn, where sign of f ‘(x) changes from +ve to –ve as the value of x increases through it. These value of x are points of local maxima. The values of the function f (x) at these points of local maxima are local maximum values of the function f (x).

(iii) Evaluate those values of x out of x1, x2, x3, ….., xn ; where sign of f ¢(x) changes from –ve to + ve as the value of x increases through it. These values of x are point of local minima.

The values of the function f (x) at these points of local minima are local minimum values of the function f (x).

4. Second-Order Derivative Test in Ascertaining the Maxima or Minima

(i) Evaluate the value of x E(a, b), where f ‘(x) = 0:

(ii) Find the value of f ¢¢ (x) at x Î (a, b), where f ‘(x) = 0:

(iii) The value of x, where f ”(x) > 0, is the point of local maxima and the value of f (x) at this value of x is called local manimum value of the function f (x).

(iv) The value of x, where f ”(x) < 0, is the point of local minima and the value of f (x) at this value of x is called local minimum value of the function f (x).

fails. In this case, f (x) can still have a maxima or minima or point of inflection (neither maxima nor minima). In this case, use the first derivative check for ascertaning the maxima or minima. If still f (x) is neither maxima nor minima, then this point is called point of inflection.

5. Approximation: Let y = f (x) be a function of x. Also suppose Del x be a small

6. Approximation of Error(s)

7. Derivatives as a Rate Measurer

8. Tangent and Normal

9. Orthogonal Curves : If the angle of intesection between two curves is a right angle, then the two curve are said to intersect orthogonally and the curves are called orthogonal curves. i.e., if m1 × m2 = –1, where m1 and m2 are slopes of tangents at the point of intersection of the two curves, then the two curves are orthogonal.

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