Question

For all x ∈ (0, 1)

• A. e^x< 1 + x
• B. log_e(1 + x) < x
• C. sin x > x
• D. log_e x > x

Answer 1

Answer: (B)

log_e(1 + x) < x

Answer 2

Let g(x) = log (1 + x) – x

⇒ g′ (x) = $\frac{1}{1 + x}$ – 1 = $\frac{–x}{1 + x}$ < 0, ∀ x ∈ (0, 1)

So, g(x) is decreasing, when 0 < x < 1

⇒ g(0) > g(x)  ⇒ log_e (1 + x) < x

Therefore, (2) is correct.