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) = x3 – 12x2 + 36x + 17

Given:- Function f(x) = x3 – 12x2 + 36x + 17


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 – 12x2 + 36x + 17


⇒


⇒ f’(x) = 3x2 – 24x + 36


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


⇒ f’(x) = 0


⇒ 3x2 – 24x + 36 = 0


⇒ 3(x2 – 8x + 12) = 0


⇒ 3(x2 – 6x – 2x + 12) = 0


⇒ x2 – 6x – 2x + 12 = 0


⇒ (x – 6)(x – 2) = 0


⇒ x = 2, 6





clearly, f’(x) > 0 if x < 2 and x > 6


and f’(x) < 0 if 2 < x < 6


Thus, f(x) increases on (–∞, 2) ∪ (6, ∞)


and f(x) is decreasing on interval x ∈ (2, 6)



Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

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