RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 21 of Exercise 17.2

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21

Prove that the function f(x) = x3 – 6x2 + 12x – 18 is increasing on R.

Given:- Function f(x) = x3 – 6x2 + 12x – 18


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) = x3 – 6x2 + 12x – 18


⇒


⇒ f’(x) = 3x2 – 12x + 12


⇒ f’(x) = 3(x2 – 4x + 4)


⇒ f’(x) = 3(x – 2)2


as given


x ϵ R


⇒ (x – 2)2> 0


⇒ 3(x – 2)2 > 0


⇒ f’(x) > 0


Hence, condition for f(x) to be increasing


Thus f(x) is increasing on interval x ∈ R


Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

More Exercise Questions

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14

Show that f(x) = tan x is an increasing function on (–π/2, π/2).
15

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19

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20

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21

Prove that the function f(x) = x3 – 6x2 + 12x – 18 is increasing on R.
22

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23

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24

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25

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26

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27

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28

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29

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30

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31

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32

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33

33 Prove that the function f(x) = cos x is :

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34

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35

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36

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37

Show that f(x) = x + cos x – a is an increasing function on R for all values of a ?
38

Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?
39

Find the intervals in which f(x) is increasing or decreasing :

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