Let f : R → R be a twice continuously differentiable function such

1.	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» 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)

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»

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)

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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) &gt; 0 for all x ≠ 0

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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