RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 7 of Exercise 17.2

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7

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π).

Given:- Function f(x) = sin 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) = sin x



f’(x) = cosx


Taking different region from 0 to 2π


a) let


cos(x) > 0


f’(x) > 0


Thus f(x) is increasing in


b) let


cos(x) < 0


f’(x) < 0


Thus f(x) is decreasing in


Therefore, from above condition we find that


f(x) is increasing in and decreasing in


Hence, condition for f(x) neither increasing nor decreasing in (0,π)


Chapter Exercises

More Exercise Questions

26

Find the interval in which is increasing or decreasing ?