Question

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

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

Answer 1

The correct option is C $f^{\prime \prime }\left(x\right)=0$ for some $x\in (0,2)$

There exists a $C_1\in (0,1)$
Such that $f^{\prime }\left(C_1\right)=\frac{f\left(1\right)-f\left(0\right)}{1-0}=1$

and there exists a $C_2\in (1,2)$
Such that $f^{\prime }\left(C_2\right)=\frac{f\left(2\right)-f\left(1\right)}{2-1}=1$

Now, by Rolle's theorem, there exists a $C\in (C_1,C_2)$ such that
$f^{\prime \prime }\left(C\right)=\frac{f^{\prime }\left(C_1\right)-f^{\prime }\left(C_2\right)}{C_1-C_2}=0$
And also $C\in (0,2)$