Find whether the following functions are increasing or decreasing or n

1.	Find whether the following functions are increasing or decreasing or neither.(i) f(x) = x4 – 8x3 + 22x2 – 24x + 5 at x = 0, -2 (ii) f(x) = 4x3 – 15x2 + 12x – 2 at x = 1,-1 (iii) f(x) = (x – 1)(x – 2)2 at x = 1,3.
Answer» (i) Given f(x) = x⁴ – 8x³ + 22x² – 24x + 5 at x = 0, -2 f'(x) = 4x³ – 24x² + 44x – 24 At x = 0, f'(0) = -24 < 0 ∴ f(x) is decreasing at x = 0 At x = -2, f'(-2) = 4(-2)³ – 24(-2)² + 44 (-2) – 24 < 0 = 32 – 96 – 88 – 24 <0 (negative) ∴ f(x) is decreasing at x = -2 (ii) Given f(x) = 4x³ – 15x² + 12x – 2 at x = 1,-1 f'(x) = 12x² – 30x + 12 At x = 1, f'(x) = 12 – 30 + 12 = -6 < 0 ∴ f(x) is increasing at x = 1 At x = -1, f'(-1) = 12(-1)² – 30 (-1) + 12 = 54 > 0 ∴ f(x) is increasing at x = -1 (iii) Given f(x) = (x – 1) (x – 2)² at x = 1, 3 f'(x) = (x – 1) (2 (x – 2) + (x – 2)²) At x = 1, f'(1) = 0 + (-1)² = 1 > 0 ∴ f(x) is increasing at x = 1 At x = 3, f'(3) = 2 × 2(1) + 1 = 5 > 0 ∴ f(x) is increasing at x = 3.

Find whether the following functions are increasing or decreasing or neither.(i) f(x) = x4 – 8x3 + 22x2 – 24x + 5 at x = 0, -2 (ii) f(x) = 4x3 – 15x2 + 12x – 2 at x = 1,-1 (iii) f(x) = (x – 1)(x – 2)2 at x = 1,3.

Answer»

(i) Given

f(x) = x⁴ – 8x³ + 22x² – 24x + 5 at x = 0, -2

f'(x) = 4x³ – 24x² + 44x – 24

At x = 0, f'(0) = -24 < 0

∴ f(x) is decreasing at x = 0

At x = -2, f'(-2) = 4(-2)³ – 24(-2)² + 44 (-2) – 24 < 0

= 32 – 96 – 88 – 24 <0 (negative)

∴ f(x) is decreasing at x = -2

(ii) Given

f(x) = 4x³ – 15x² + 12x – 2 at x = 1,-1

f'(x) = 12x² – 30x + 12

At x = 1, f'(x) = 12 – 30 + 12 = -6 < 0

∴ f(x) is increasing at x = 1

At x = -1, f'(-1) = 12(-1)² – 30 (-1) + 12 = 54 > 0

∴ f(x) is increasing at x = -1

(iii) Given

f(x) = (x – 1) (x – 2)² at x = 1, 3

f'(x) = (x – 1) (2 (x – 2) + (x – 2)²)

At x = 1, f'(1) = 0 + (-1)² = 1 > 0

∴ f(x) is increasing at x = 1

At x = 3, f'(3) = 2 × 2(1) + 1 = 5 > 0

∴ f(x) is increasing at x = 3.