## 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) = 2x3 – 24x + 7

Given:- Function f(x) = 2x3 – 24x + 7

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 – 24x + 7

f’(x) = 6x2 – 24

For f(x) to be increasing, we must have

f’(x) > 0

6x2 – 24 > 0

x2 < 4

x < –2, +2

x (–,–2) and x (2,∞)

Thus f(x) is increasing on interval (–, –2) (2, ∞)

Again, For f(x) to be increasing, we must have

f’(x) < 0

6x2 – 24< 0

x2 < 4

x> –1

x (–1,∞)

Thus f(x) is decreasing on interval x (–1, ∞)

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