RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 5 of Exercise 17.2

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5

Show that , x ≠ 0 is a decreasing function for all x ≠ 0.

Given:- Function


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,



⇒


⇒


⇒


As given x ∈ R , x ≠ 0


⇒


Their ratio is also greater than 0


⇒


⇒ ; as by applying –ve sign change in comparision sign


⇒ f’(x) < 0


Hence, condition for f(x) to be decreasing


Thus f(x) is decreasing for all x ≠ 0


Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

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