Question

The values of k for which the function f(x) = kx³ – 6x² + 12x + 11 may be increasing on R are ______.

विकल्प

• k > 1

• k < 1

• k ≤ 1

• none of these

Answer 1

The values of k for which the function f(x) = kx³ – 6x² + 12x + 11 may be increasing on R are k > 1.

Explanation:

f'(x) =  3kx² – 12x + 12

= 3(kx² – 4x + 4)

Since f(x) is increasing on R, f'(x) > 0

∴ kx² – 4x + 4 > 0 ∀ x ∈ R

⇒ k > 0 and 16 – 16k < 0  ......[∵ ax² +bx + c > 0 ∀ x ∈ R ⇒ a > 0 and b² – 4ac < 0]

⇒ k > 1

Hence, f(x) is increasing on R if k > I.