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Chapter: 12th Maths : UNIT 7 : Applications of Differential Calculus

Exercise 7.6 : Applications of First Derivative

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EXERCISE 7.6

 

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

(i) ( x ) = x 2 12+10 ; [1, 2]

(i) ( 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 = 3π / 4, 7π / 4.

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