## 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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##### Find the intervals in which the following functions are increasing or decreasing.f(x) = x4 – 4x3 + 4x2 + 15

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,

f’(x) = 4x3 – 12x2 + 8x

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

f’(x) = 0

4x3 – 12x2 + 8x= 0

4(x3 – 3x2 + 2x) = 0

x(x2 – 3x + 2) = 0

x(x2 – 2x – x + 2) = 0

x(x – 2)(x – 1)

x = 0, 1 , 2

clearly, f’(x) > 0 if 0 < x < 1 and x > 2

and f’(x) < 0 if x < 0 and 1 < x < 2

Thus, f(x) increases on (0, 1) (2, ∞)

and f(x) is decreasing on interval (–∞, 0) (1, 2)

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