Applications of Derivatives Class 12 Notes Maths Chapter 6 - CBSE
Chapter:6
What are Applications of Derivatives ?
Rate Of Change Of Bodies
If a quantity y varies with another quantity x, satisfying some rule y = f(x), then dy/dx represents the rate of change of y w.r.t x and dy/dx x=x0 represents the rate of change of y w.r.t x at x=x0.
Increasing/decreasing Functions
A function f is said to be :
- Increasing on an interval (a,b) if x1 < x2 in (a,b) ⇒ f (x1) ≤ f (x2) for all x1, x2 ∈(a,b).
- Decreasing on an interval (a,b) if x1 < x2 in (a,b) ⇒ f (x1) ≥ f (x2) for all x1, x2 ∈(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.
