Question

Let f : R → R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then

(A) f''(0) = 0

(B) f''(c) = 0 for some c ∈ R

(C) if c ≠ 0, then f ''(c) ≠ 0

(D) f'(x) > 0 for all x ≠ 0

Answer 1

The correct option (B) f''(c) = 0 for some c ∈ R

Explanation:

f(x) is continuous and differentiable 

f(0) = f(1) = 0 ⇒ by rolles theorem 

f'(a) = 0 , a ∈ (0, 1) 

given f'(0) = 0 by rolles theorem 

f''(0) = 0 for some c, c ∈ (0, a)