FIND ABSOLUTE EXTREMA OF FUNCTION ON CLOSED INTERVAL

The absolute maxima and absolute minima are referred to describing the largest and smallest values of a function on an interval.

Let x₀ be a number in the domain D of a function f(x) . Then f (x₀) is the absolute maximum value of f (x) on D,

if f(x₀) ≥ f(x) for all x ∈ D and f(x₀) is the absolute minimum value of f(x) on D if f(x₀) ≤ f(x) 

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

(i) f (x)  =  x³ − 12x + 10 ; [1, 2]

Solution :

f (x)  =  x³ − 12x + 10

f'(x)  =  3x²-12(1)

f'(x)  =  3x²-12

f'(x)  =  0

 3x²-12  =  0

x²  =  4

x  =  ±2

x = 2 ∈ [1, 2]

f(1)  =  1³ − 12(1) + 10  ==>  -1

f(2)  =  2³ − 12(2) + 10

=  8-24+10

=  -6

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

(ii)  f(x)  =  3x⁴-4x³ ; [-1, 2]

Solution :

f(x)  =  3x⁴-4x³

f'(x)  =  12x³ - 12x²

f'(x)  =  0

12x³ - 12x²  =  0

12x²(x-1)  =  0

x  =  0 and x  =  1

f(x)  =  3x⁴-4x³

f(-1)  =  3(-1)⁴-4(-1)³  =  7

f(0)  =  3(0)⁴-4(0)³  =  0

f(1)  =  3(1)⁴-4(1)³  =  -1

f(2)  =  3(2)⁴-4(2)³  =  16

So, absolute maximum is 16 and absolute minimum is -1.

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

Solution :

f(x)  =  6x^4/3 - 3x^1/3

f'(x)  =  6 (4/3) x^1/3  - 3(1/3)x^-2/3

f'(x)  =  8x^1/3  - x^-2/3

f'(x)  =  0

8x^1/3  - x^-2/3  =  0

8x^1/3  =  x^-2/3 

x^1/3  x^2/3  =  1/8

x  =  1/8

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

Solution :

f(x)  =  2 cos x + sin 2x

f'(x)  =  -2 sinx + 2 cos 2x

f'(x)  =  0

-2 sinx + 2 cos 2x  =  0

-2 sinx + 2 (1-2sin²x)  =  0

-2 sinx + 2 - 4sin²x  =  0

4 sin²x+2sinx-2  =  0

Let t  =  sinx

4t²+2t-2  =  0

2t²+t-1  =  0

(2t-1) (t+1)  =  0

t  =  1/2 and t  =  -1

sin x  =  1/2 and sin x  =  -1

f(x)  =  2 cos x + sin 2x

f(π/2)  =  2 cos π/2 + sin 2(π/2)

f(π/2)  =  0

f(0)  =  2 cos 0 + sin 2(0)

f(0)  =  2

f(π/6)  =  2 cos π/6 + sin 2(π/6)

f(π/2)  =  1+√3/2

So, absolute maximum is 2 and absolute minimum is 0.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.