Question

Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval ______. 

Options

• [15, 19)

• (– ∞, 12)

• [12, 15)

• [19, ∞)

Answer 1

Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval [19, ∞). 

Explanation:

Given f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6

Consider f'(x) = `(f(x + h) - f(x))/h`

⇒ f(x + h) – f(x) = f'(x) . h ≥ (4.2)h

So, f(x + h) ≥ f(x) + (4.2)h

Put x = 1 and h = 5, we get

f(6) ≥ f(1) + 5(4.2)

⇒ f(6) ≥ 19

Hence f(6) lies in [19, ∞)