Let f be any continuous function on [0,2] and twice differentiable on (0,2). If f (0)=0, f (1)=1 and f(2)=2, then

Question

Let f be any continuous function on [0,2] and twice differentiable on (0,2). If f (0)=0, f (1)=1 and f(2)=2, then (A) f prime prime(x)=0 for all x ∈(0,2) (B) f prime prime(x)=0 for some x ∈(0,2) (C) f prime(x)=0 for some x ∈[0,2] (D) f prime prime(x)>0 for all x ∈(0,2)

Tardigrade Admin · Accepted Answer

f(0)=0 f(1)=1 and f(2)=2 Let h(x)=f(x)-x has three roots By Rolle's theorem h' (x)=f prime(x)-1 has at least two roots h" (x)=f prime prime(x)=0 has at least one roots

Q. Let $f$ be any continuous function on $[0, 2]$ and twice differentiable on $(0, 2)$ . If $f (0) = 0, f (1) = 1$ and $f (2) = 2$ , then

$f^{''} (x) = 0$ for all $x \in (0, 2)$

$f^{''} (x) = 0$ for some $x \in (0, 2)$

$f^{'} (x) = 0$ for some $x \in [0, 2]$

$f^{''} (x) > 0$ for all $x \in (0, 2)$

$f (0) = 0 f (1) = 1$ and $f (2) = 2$
Let $h (x) = f (x) - x$ has three roots
By Rolle's theorem $h^{'}$ $(x) = f^{'} (x) - 1$ has at least two roots $h "$ $(x) = f^{''} (x) = 0$ has at least one roots

Q. Let f be any continuous function on [0,2] and twice differentiable on (0,2). If f(0)=0,f(1)=1 and f(2)=2, then

f′′(x)=0 for all x∈(0,2)

f′′(x)=0 for some x∈(0,2)

f′(x)=0 for some x∈[0,2]

f′′(x)>0 for all x∈(0,2)