## 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

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