Question

Let f(x) is differentiable function in [0, 2]. f(0) = 0 and

f ′(x) ≤$\frac{1}{2}$∀ x∈[0, 2], then

• A. |f(x)| ≤ 2
• B. f(x) ≤ 1
• C. f(x) = 2x
• D. f(x) = 3 for some x ∈ (0, 2)

Answer 1

Answer: (B)

f(x) ≤ 1

Answer 2

Using LMVT

f ′(3) = $\frac{f(x) - f(0)}{x - 0}$≤ $\frac{1}{2}$

⇒ f(x) ≤ $\frac{x}{2}$ ∀ x∈(0, 2)