Question

Let f be twice differentiable function satisfying f(1) = 1, f(2) = 4, f(3) = 9, then

(a) f"(x) = 2, ∀ x ∈ (R)

(b) f'(x) = 5 = f "(x), for some x ∈ (1, 3)

(c) there exists at least one x ∈ (1, 3) such that f"(x) = 2 

(d) None of the above

Answer 1

Correct option (c) there exists at least one x ∈ (1, 3) such that f"(x) = 2 

Explanation :

Let, g(x) = f(x) - x² 

g(x ) has at least 3 real roots which are x = 1, 2, 3 (by mean value theorem)

g(x) has at least 2 real roots in x ∈ (1, 3)

g "(x) has at least 1 real roots in x ∈ (1, 3)

f "(x) - 2.1 = 0 for at least 1 real root in x ∈ (1, 3)

f ''(x) = 2, for at least one root in x ∈ (1, 3 )