Exercise 7.6 : Applications of First Derivative

EXERCISE 7.6

1. Find the absolute extrema of the following functions on the given closed interval.

(i) f ( x ) = x 2 âˆ’12x +10 ; [1, 2]

(i) f ( x ) = 3x4 âˆ’ 4x3 ; [−1, 2]

(i) f (x) = 6x4/3 âˆ’3x1/3 ; [−1, 1]

(iv) f (x) = 2 cos x + sin 2x ; [0, Ï€/2]

2. Find the intervals of monotonicities and hence find the local extremum for the following functions:

(i) f (x) = 2x3 + 3x2 âˆ’12x  

(ii) f (x) = x / x âˆ’ 5

(iii) f (x) = ex / 1-x3

(iv) f (x) = ex/3 − log x

(v) f (x) = sin x cos x + 5, x âˆˆ(0, 2Ï€ )

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 = Ï€ / 4, 5Ï€ /4.

local minimum= 9/2 at x = 3Ï€ / 4, 7Ï€ / 4.