1. Given data

We have a function $f$ such that:

• $f$ is continuous on $[0,2]$
• $f$ is twice differentiable on $(0,2)$
• $f(0)=0,\, f(1)=1,\, f(2)=2$

We must determine which option is necessarily true.

2. Observe the three given points

The points are:

These all lie on the straight line

So, define

Then:

Thus $g$ is continuous on $[0,2]$, twice differentiable on $(0,2)$, and has zeros at $0,1,2$.

3. Apply Rolle's theorem twice

Since $g(0)=g(1)=0$, by Rolle's theorem there exists $c_1\in(0,1)$ such that

Similarly, since $g(1)=g(2)=0$, there exists $c_2\in(1,2)$ such that

Now $g'$ is continuous on $[c_1,c_2]$ and differentiable on $(c_1,c_2)$, with

Applying Rolle's theorem again to $g'$, there exists some $c\in(c_1,c_2)\subset(0,2)$ such that

But

since the second derivative of $x$ is $0$. Hence,

for some $c\in(0,2)$.

Therefore, option B is true.

4. Check the other options

Option A: $f''(x)=0$ for all $x\in(0,2)$

This would mean $f$ is linear on $(0,2)$. It is not necessary that every such function be linear.

Example:

for any nonzero constant $a$. Then

but $f''(x)$ is not identically zero. So A is false.

Option C: $f'(x)=0$ for some $x\in[0,2]$

Not necessary. For example,

satisfies all conditions, but

for all $x$, so $f'(x)$ is never $0$. Thus C is false.

Option D: $f''(x)>0$ for all $x\in(0,2)$

If $f''(x)>0$ for all $x$, then $f''$ can never be $0$, which contradicts what we proved in Step 3. So D is false.

5. Conclusion

The only necessarily true statement is: