Question

Let f be a real valued function defined on the interval (0, ∞) by

f(x) = lnx + ∫(1 + sintdt) for t ∈ [0, x]

Then which one of the following statements is true.

(a) f ≤ (x) exists for all x ∈ (0, ∞)

(b) f'(x) exists for all x ∈(0, ∞) and and f¢ is continuous on (0, ∞) but not differentiable on (0, ∞)

(c) there exists a > 1 such that |f'(x)| < |f(x)| for all x ∈ (a, ∞)

(d) there exists b > 0 such that |f'(x)| + |f(x)| ≤ b for all x ∈(0, ∞) 

Answer 1

Correct option (b, c)

Explanation:

Clearly, f'(x) exists for all x > 0

Thus, f'(x) is continuous for all x > 0 but not differentiable at x > 0.

More-over, f'(x), f(x) > 0 " x ∈ (1, ∞)

Thus, 1/x is not bounded.