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

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

Tardigrade Admin · Accepted Answer

We have, f: R arrow R be a twice continuously differentiable function such that f(0)=f(1)=f prime(0)=0 Now, for atleast one value of c1 ∈(0,1), f prime(c1)=0 (by Rolle's theorem) Again, f prime(0)=0=f prime(c1) ⇒ f prime(c)=0 for some c ∈(0, c1) (by Rolle's theorem)

Q. Let $f : R \to R$ be a twice continuously differentiable function such that $f (0) = f (1) = f^{'} (0) = 0$ . Then

$f^{''} (0) = 0$

$f^{''} (c) = 0$ for some $c \in R$

if $c \neq = 0$ , then $f^{''} (c) \neq = 0$

$f^{'} (x) > 0$ for all $x \neq = 0$

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

f′′(0)=0

f′′(c)=0 for some c∈R

if c=0, then f′′(c)=0

f′(x)>0 for all x=0

We have, f:R→R be a twice continuously differentiable function such that f(0)=f(1)=f′(0)=0 Now, for atleast one value of c1​∈(0,1), f′(c1​)=0 (by Rolle's theorem) Again, f′(0)=0=f′(c1​) ⇒f′(c)=0 for some c∈(0,c1​) (by Rolle's theorem)

Q. Let $f : R \to R$ be a twice continuously differentiable function such that $f (0) = f (1) = f^{'} (0) = 0$ . Then

$f^{''} (0) = 0$

$f^{''} (c) = 0$ for some $c \in R$

if $c \neq = 0$ , then $f^{''} (c) \neq = 0$

$f^{'} (x) > 0$ for all $x \neq = 0$