If f(x) = kx3 - 9x2+9x+3 is increasing for every real number x, thenA.

1.	If f(x) = kx3 - 9x2+9x+3 is increasing for every real number x, thenA. k > 3B. k ≥ 3C. k < 3D. k ≤ 3
Answer» Answer is : A. k > 3 Given f(x) = kx³ - 9x²+ 9x+3 ⇒ f’(x) = 3kx²- 18x+9 ⇒ f’(x) = 3(kx² – 6x + 3) > 0 ⇒ kx² – 6x + 3 > 0 For quadratic equation to be greater than 0. a > 0 and D < 0 ⇒ k > 0 and (-6)²- 4(k)(3) < 0 ⇒ 36 – 12k < 0 ⇒ 12k > 36 ⇒ k > 3 ∴ k > 3.

If f(x) = kx3 - 9x2+9x+3 is increasing for every real number x, thenA. k > 3B. k ≥ 3C. k < 3D. k ≤ 3

Answer»

Answer is : A. k > 3

Given f(x) = kx³ - 9x²+ 9x+3

⇒ f’(x) = 3kx²- 18x+9

⇒ f’(x) = 3(kx² – 6x + 3) > 0

⇒ kx² – 6x + 3 > 0

For quadratic equation to be greater than 0. a > 0 and D < 0

⇒ k > 0 and (-6)²- 4(k)(3) < 0

⇒ 36 – 12k < 0

⇒ 12k > 36

⇒ k > 3

∴ k > 3.