Question

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

A. k > 3

B. k ≥ 3

C. k < 3

D. k ≤ 3

Answer 1

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.