EXERCISE 7.6
1. Find the absolute extrema of the following functions on the given closed interval.
(i) f ( x ) = x 2 −12x +10 ; [1, 2]
(i) f ( x ) = 3x4 − 4x3 ; [−1, 2]
(i) f (x) = 6x4/3 −3x1/3 ; [−1, 1]
(iv) f (x) = 2 cos x + sin 2x ; [0, π/2]
2. Find the intervals of monotonicities and hence find the local extremum for the following functions:
(i) f (x) = 2x3 + 3x2 −12x  
(ii) f (x) = x / x − 5
(iii) f (x) = ex / 1-x3
(iv) f (x) = ex/3 − log x
(v) f (x) = sin x cos x + 5, x ∈(0, 2π )
Answers:
(1)
(i) absolute maximum = −1 , absolute minimum = −26
(ii) absolute maximum = 16 , absolute minimum = −1
(iii) absolute maximum = 9 , absolute minimum = − 9/8
(iv) absolute maximum = 3√3 / 2 , absolute minimum = 0
(2)
(i) strictly increasing on ( -∞, 2 ) and (1,∞) , strictly decreasing on (−2,1)
local maximum = 20
local minimum = −7
(ii) strictly decreasing on ( −∞, 5) and (5, ∞) . No local extremum.
(iii) strictly increasing on ( −∞, ∞). No local extremum.
(iv) strictly decreasing on ( 0,1) , strictly increasing on (1, ∞). local minimum = 1/3
(v) strictly increasing on
strictly decreasing on , . local maximum= 11/2 at x = π / 4, 5π /4.
local minimum= 9/2 at x = 3Ï€ / 4, 7Ï€ / 4.
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