# Applications of Derivatives Class 12 Notes Maths Chapter 6 - CBSE

## Chapter:6

## What are Applications of Derivatives ?

## Rate Of Change Of Bodies

_{0} represents the rate of change of y w.r.t x at x=x_{0}.

## Increasing/decreasing Functions

A function f is said to be :

- Increasing on an interval (a,b) if x
_{1}< x_{2}in (a,b) ⇒ f (x_{1}) ≤ f (x_{2}) for all x_{1}, x_{2}**∈**(a,b). - Decreasing on an interval (a,b) if x
_{1}< x_{2}in (a,b) ⇒ f (x_{1}) ≥ f (x_{2}) for all x_{1}, x_{2}**∈**(a,b).

## Maxima And Minima

**First Derivative Test**

Let f be a function defined on an open interval I. Let f be continuous at a critical point a in I. Then

- If
**f '(x)**> 0 at every point sufficiently close to and to the left of a, and**f '(x)**< 0 at every point sufficiently close to and to the right of a, then a is a point of**local maxima.** - If
**f '(x)**< 0 at every point sufficiently close to and to the left of a, and**f '(x)**> 0 at every point sufficiently close to and to the right of a, then a is a point of**local minima.** - If
**f '(x)**does not change sign as x increases through a, then a is neither a point of local maxima nor a point of local minima. Infact, such a point is called**point of inflexion.**

**Second Derivative Test**

Let** f** be a function defined on an interval I and a **∈** I. Let f be twice differential at c. Then

- x = a is a point of local maxima if
**f '(a)**= 0 and**f ''(a) < 0**The values f(a) is local maximum value of f. - x = a is a point of local minima if
**f '(a)**= 0 and**f ''(a) > 0**In this case, f(a) is local minimum value of f. - The test fails if
**f '(a)**= 0 and**f ''(a) = 0**In this case, we go back to the first derivative test and find whether a 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 f in the interval, i.e., find points x where either f '(x) = 0 or f 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 f. - ➥
**Step: 4**Identify the maximum and minimum values if f out of the values calculated in step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f.