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f '' ( x) =- sin x - 1 (cos 2x) × 2 =- sin x - 2 cos 2x

2

As f ''  π  =- sin π - 2 cosπ =-1 - 2 × (-1) = 2 -1 > 0
 2  2

and f ''  3π  =- sin 3π - 2 cos 3π =-(-1) - 2 (-1) =1+ 2 >0
 2  2

Thus f ( x) has minimum value=s for x π=and x 3π

22

As f ''  π  =- sin π - 2 cos π =- 1 - 2 . 0 =- 1 < 0
 4  4 22 2

and f ''  3π  =- sin 3π - 2 cos 3π =- 1 - 2 . 0 =- 1 < 0
 4  4 2 2 2

Thus f ( x) has minimum value=s for x π=and x 3π

44

EXERCISE 2.9

1. Determine the intervals in which f is increasing or decreasing for the domain
mentioned in each case.

(i) f ( x) = sin x ; x ∈(-π ,π )

(ii) f ( x) = cos x ; x ∈  -π ,π 
 2 2 
(iii) f ( x)= 4 - x2 ; x ∈(-2,2)

(iv) f ( x) = x2 + 3x + 2 ; x ∈(-4,1)

2. Find the extreme values for the following functions defined as:

(i) f ( x)= 1- x3 (ii) f ( x) = x2 - x - 2
(iii) f ( x) = 5x2 - 6x + 2 (iv) f ( x) = 3x2
(v) f ( x) = 3x2 - 4x + 5 (vi) f ( x) = 2x3 - 2x2 - 36x + 3

version: 1.1

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