RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 1 of Exercise 17.2

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1

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x3 + 9x2 + 12x + 20

Given:- Function f(x) = 2x3 + 9x2 + 12x + 20


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) = 2x3 + 9x2 + 12x + 20


⇒


⇒ f’(x) = 6x2 + 18x + 12


For f(x) lets find critical point, we must have


⇒ f’(x) = 0


⇒ 6x2 + 18x + 12 = 0


⇒ 6(x2 + 3x + 2) = 0


⇒ 6(x2 + 2x + x + 2) = 0


⇒ x2 + 2x + x + 2 = 0


⇒ (x + 2)(x + 1) = 0


⇒ x = –1, –2





clearly, f’(x) > 0 if –2 < x < –1


and f’(x) < 0 if x < –1 and x > –2


Thus, f(x) increases on x ∈ (–2,–1)


and f(x) is decreasing on interval (–∞, –2) ∪ (–2, ∞)



Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

More Exercise Questions

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14

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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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36

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37

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38

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39

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