Test whether the following function are increasing or decreasing. i. f(x) = x3 + 10x + 7 for x r

Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.

f(x) = x3 – 6x2 + 12x – 16

∴ f'(x) = `"d"/"dx" ("x"^3 - "6x"^2 + "12x" - 16)`

= 3x2 – 6 × 2x + 12 × 1 – 0
= 3x2 – 12x + 12
= 3(x2 – 4x + 4)
= 3(x - 2)2 ≥ 0 for all x ∈ R∴ f'(x) ≥ 0 for all x ∈ R

∴ f is increasing for all x ∈ R.

Concept: Increasing and Decreasing Functions

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Page 2

Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.

f(x) = 2 – 3x + 3x2 – x3 

∴ f'(x) = `d/dx(2 - 3x + 3x^2 - x^3)`

= 0 – 3 x 1 + 3 x 2x – 3x2
= – 3 + 6x – 3x2
= –3(x2 – 2x + 1)
= – 3(x – 1)2 ≤ 0 for all x ∈ R∴ f'(x) ≤ 0 for all x ∈ R

∴ f is decreasing for all x ∈ R.

Concept: Increasing and Decreasing Functions

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Page 3

Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.

f(x) = `(1)/x`

∴ f'(x) = `d/dx(x - 1/x)`

= `1 - ((-1)/x^2)`

= `1 + (1)/x^2` > 0 for all x ∈ R , x ≠ 0

∴ f'(x) > 0 for all x ∈ R, where x ≠ 0
∴ f is increasing for all x ∈ R, where x ≠ 0.

Concept: Increasing and Decreasing Functions

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Page 4

Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6

f(x) = 2x3 – 3x2 – 12x + 6

∴ f'(x) = `d/dx(2x^3 - 3x^2 - 12x + 6)`

= 2 x 3x2 – 3 x 2x – 12 x 1 + 0
= 6x2 – 6x – 12
= 6(x2 – x – 2)f is strictly increasing if f'(x) > 0

i.e. if 6(x2 – x – 2) > 0

i.e. if `x^2 - x + (1)/(4) > 2 + (1)/(4)`

i.e. if `(x - 1/2)^2 > (9)/(4)`

i.e. if `x - (1)/(2) > (3)/(2) or x - (1)/(2) < - (3)/(2)`

i.e. if x > 2 or x < – 1
∴ f is strictly increasing if x < – 1 or x > 2.

Concept: Increasing and Decreasing Functions

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Page 5

Find the values of x for which the following functions are strictly increasing : f(x) = 3 + 3x – 3x2 + x3 

f(x) = 3 + 3x – 3x2 + x3 

∴ f'(x) = `d/dx(3 + 3x - 3x^2 + x^3)`

= 0 + 3 x 1 – 3 x 2x + 3x2
= 3 – 6x + 3x2
= 3(x2 – 2x + 1)f is strictly increasing if f'(x) > 0

i.e. if 3(x2 – 2x  + 1) > 0

∴ f is strictly increasing if x ∈ R – { 1 }.

Concept: Increasing and Decreasing Functions

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Page 6

Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7

f(x) = x3 – 6x2 – 36x + 7

∴ f'(x) = `d/dx(x^3 - 6x^2 - 36x + 7)`

= 3x2 – 6 x 2x – 36 x 1 + 0
= 3x2 – 12x – 36
= 3(x2 – 4x – 12)f is strictly increasing if f'(x) > 0

i.e. if 3(x2 – 4x – 12) > 0

∴  f is strictly increasing if x < – 2 or x > 6.

Concept: Increasing and Decreasing Functions

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Page 7

Find the values of x for which the following functions are strictly decreasing : f(x) = 2x3 – 3x2 – 12x + 6

f(x) = 2x3 – 3x2 – 12x + 6

∴ f'(x) = `d/dx(2x^3 - 3x^2 - 12x + 6)`

= 2 x 3x2 – 3 x 2x – 12 x 1 + 0
= 6x2 – 6x – 12
= 6(x2 – x – 2)f is strictly decreasing if f'(x) < 0

i.e. if 6(x2 – x – 2) < 0

i.e. if `x^2 - x + (1)/(4) < 2 + (1)/(4)`

i.e. if `(x - 1/2)^2 < (9)/(4)`

i.e. if `-(3)/(2) < x - (1)/(2) < (3)/(2)`

i.e. if `-(3)/(2) + (1)/(2) < x -(1)/(2) + (1)/(2) < (3)/(2) + (1)/(2)`

i.e. if – 1 < x < 2
∴ f is strictly decreasing if – 1 < x < 2.

Concept: Increasing and Decreasing Functions

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Page 8

Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`

f(x) = `x + (25)/x`

∴ f'(x) = `d/dx(x + 25/x)`

= 1 + 25 (– 1)x–2

= `1 - (25)/x^2`

f is strictly decreasing if f'(x) < 0

i.e. if `1 - (25)/x^2 < 0`

i.e. if  `1 < (25)/x^2`

i.e. if x2 < 25i.e. if –5 < x < 5, x ≠ 0i.e. if x ∈ (– 5, 5) – { 0 }

∴ f is strictly decreasing if x ∈ (– 5, 5) – { 0 }.

Concept: Increasing and Decreasing Functions

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Page 9

f(x) = x3 – 9x2 + 24x + 12

∴ f'(x) = `d/dx(x^3 - 9x^2 + 24x + 12)`

= 3x2 – 9 x 2x + 24 x 1 + 0
= 3x2 – 18x + 24
= 3(x2 – 6x + 8)f is strictly decreasing if f'(x) < 0

i.e. if 3(x2 – 6x + 8) < 0

∴ f is strictly decreasing if x ∈ (2, 4).

Page 10

f(x) = x3 – 12x2 – 144x + 13

∴ f'(x) = `d/dx(x^3 - 12x^2 - 144x + 13)`

= 3x2 – 12 x 2x – 144 x 1 + 0
= 3x2 – 24x – 144
= 3(x2 – 8x – 48)

(a) if is increasing if f'(x) ≥ 0
i.e. if 3(x2 – 8x – 48) ≥ 0
i.e. if x2 – 8x – 48 ≥ 0
i.e. if x2 – 8x ≥ 48
i.e. if x2 – 8x + 16 ≥ 48 + 16
i.e. if (x – 4)2 ≥ 64i.e. if x – 4 ≥ 8 or x – 4 ≤ – 8i.e. if x ≥ 12 or x ≤ – 4∴ f is increasing if x ≤ – 4 or x ≥ 12,

i.e. x ∈ `( - oo, - 4] ∪ [12, oo)`.

(b) f is decreasing if f'(x) ≤ 0
i.e. if 3(x2 – 8x – 48) ≤ 0
i.e. if x2 – 8x – 48 ≤ 0
i.e. if x2 – 8x ≤ 48 
i.e. if x2 – 8x + 16 ≤ 48 + 16
i.e. if (x – 4)2 ≤ 64i.e. if – 8 ≤ x – 4 ≤  8i.e. if – 4 ≤ x ≤ 12

∴ f is decreasing if – 4 ≤ x ≤ 12, i.e. x ∈[– 4, 12].

Page 11

f(x) = 2x3 – 15x2 – 144x – 7

∴ f'(x) = `"d"/("d"x)(2x^3 - 15x^2 - 144x - 7)`

= 2 × 3x2 – 15 × 2x – 144 × 1 – 0

= 6x2 – 30x – 144

= 6(x2 – 5x – 24)

(a) f(x) is strictly increasing if f'(x) > 0

i.e. if 6(x2 – 5x – 24) > 0

i.e. if x2 – 5x –24 > 0

i.e. if x2 – 5x > 24

i.e. if `x^2 - 5x + (25)/(4) > 24 + (25)/(4)`

i.e. if `(x - 5/2)^2 > (121)/(4)`

i.e. if `x - (5)/(2) > (11)/(2) or x - (5)/(2) < - (11)/(2)`

i.e. if x > 8 or x < – 3

∴ f(x) is strictly increasing, if x < – 3 or x > 8.

(b) f(x) is strictly decreasing if f'(x) < 0

i.e. if 6(x2 – 5x – 24) < 0

i.e. if x2 – 5x –24 < 0

i.e. if x2 – 5x < 24

i.e. if `x^2 - 5x + (25)/(4) < 24 + (25)/(4)`

i.e. if `(x - 5/2)^2 < (121)/(4)`

i.e. if `x - (5)/(2) < (11)/(2) or x - (5)/(2) < - (11)/(2)`

i.e. if `-(11)/(2) + (5)/(2) < x - (5)/(2) + (5)/(2) < (11)/(2) + (5)/(2)`

i.e. if – 3 < x < 8

∴ f(x) is strictly decreasing, if – 3 < x < 8.

Page 12

f(x) = `x/(x^2 + 1)`

∴ f'(x) = `d/dx(x/(x^2 + 1))`

= `((x^2 + 1).d/dx(x) - xd/dx(x^2 + 1))/(x^2 + 1)^2`

= `((x^2 + 1)(1) - x(2x + 0))/(x^2 + 1)^2`

= `(x^2 + 1 - 2x^2)/(x^2 + 1)^2`

= `(1 - x^2)/(x^2 + 1)^2`

(a) f is strictly increasing if f'(x) > 0

i.e. if `(1 - x^2)/(x^2 + 1)^2 > 0`

i.e. if 1 – x2 > 0             ...[∵ (x2 + 1)2 > 0]
i.e. if 1 > x2
i.e. if x2 < 1i.e. if – 1 < x < 1

∴ f is strictly increasing if – 1 < x < 1

(b) f is strictly increasing if f'(x) < 0

i.e. if `(1 - x^2)/(x^2 + 1)^2 < 0`

i.e. if 1 – x2 < 0             ...[∵ (x2 + 1)2 > 0]
i.e. if 1 < x2
i.e. if x2 > 1i.e. if  x > 1 or x < – 1∴ f is strictly increasing if x < – 1 or x > 1

i.e. `x ∈( - oo, - 1) ∪ (1, oo)`.