RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 13 of Exercise 17.2

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13

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (–π, 0) and neither increasing nor decreasing in (–π, π).

Given:- Function f(x) = cos x


Theorem:- Let f be a differentiable real function defined on an open interval (a,b).


(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)


(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)


Algorithm:-


(i) Obtain the function and put it equal to f(x)


(ii) Find f’(x)


(iii) Put f’(x) > 0 and solve this inequation.


For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.


Here we have,


f(x) = cos x



f’(x) = –sinx


Taking different region from 0 to 2π


a) let


sin(x) > 0


–sinx < 0


f’(x) < 0


Thus f(x) is decreasing in


b) let


sin(x) < 0


–sinx > 0


f’(x) > 0


Thus f(x) is increasing in


Therefore, from above condition we find that


f(x) is decreasing in and increasing in


Hence, condition for f(x) neither increasing nor decreasing in (–π,π)


13

Chapter Exercises

More Exercise Questions

26

Find the interval in which is increasing or decreasing ?