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 $x \in(0,2)$
• B. $f^{\prime \prime}(x)=0$ for some $x \in(0,2)$
• C. $f^{\prime}(x)=0$ for some $x \in[0,2]$
• D. $f^{\prime \prime}(x)>0$ for all $x \in(0,2)$

Answer 1

Answer: (B)

$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