Question

There exists a function f(x) satisfying f(0) = 1, f'(0) = -1, f(x) > 0 for all x and

(a) f"(x) < 0 for all x

(b) -1 < f:(x) < 0 for all x

(c) -2 ≤ f"(x) ≤ -1 for all x

(d) f"(x) < -2 for all x

Answer 1

Correct option (a) f"(x) < 0 for all x

Explanation :

Since, f(x) is continuous and differentiable where, f(0) = 1 and f'(0) = - 1, f(x) > 0 for all x.

Thus, f(x) is decreasing for x > 0 and concave down.

f"(x) < 0

Therefore, (a) is answer.