Question

Let ƒ(x) satisfy the requirements of Lagrange’s mean value theorem in [0, 2]. If ƒ(0) = 0 and ƒ ¢ (x) £$\frac{1}{2}$ for all x in [0, 2], then –

• A. | ƒ(x) | £ 2
• B. ƒ(x) £ 1
• C. ƒ(x) = 2x
• D. ƒ(x) = 3 for at least one x in [0, 2]

Answer 1

Answer: (B)

ƒ(x) £ 1

Answer 2

Applying Lagrange’s mean value theorem to ƒ (x) in [0, x], 0 < x £ 2, we get $\frac{ƒ(x) - ƒ(0)}{x - 0}$ = ƒ¢ (3) for some c Ī (0, 2)

Ž $\frac{ƒ(x)}{x}$ = ƒ ¢ (3) £ $\frac{1}{2}$

Ž ƒ (x) £ $\frac{1}{2}$ x £ $\frac{1}{2}$ . 2 (Q x £ 2)

Ž ƒ (x) £ 1.