APPLICATIO OF FIRST DERIVATIVES PRACTICE QUESTIONS

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

Problem 1 :

f (x)  =  x³ − 12x + 10 ; [1, 2]

Problem 2 :

f(x)  =  3x⁴ - 4x³ ; [-1, 2]

Problem 3 :

f(x)  =  6x^4/3 - 3x^1/3 ;  [-1, 1]

1)  Absolute maximum is -1 and absolute minimum is -6.

2)   absolute maximum is 16 and absolute minimum is -1.

3) So, relative minimum is at x  1/8.

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

(i) f (x)  =  x³ − 12x + 10 ; [1, 2]        Solution

(ii)  f(x)  =  3x⁴-4x³ ; [-1, 2]       Solution

(iii)  f(x)  =  6x^4/3 - 3x^1/3 ;  [-1, 1]       Solution

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

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

(i)  f(x) = 2x³ + 3x² −12x         Solution

(ii)  f(x)  =  x/(x-5)             Solution

(iii)  f(x)  =  e^x/(1-e^x)           Solution

(iv)  f(x)  =  (x³/3) - log x         Solution

(v)  f(x)  =  sinx cosx + 5, x ∈ (0, 2π)        Solution

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