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Suppose that $g(x)$ is a twice differentiable function and $g(1) = 1$; $g(2) = 4$; $g(3) = 9$. Which of the following is necessarily true?

(A) $g''(x) = 3$ for some $x$ belonging to $[1, 2]$

(B) $g''(x) = 5$ for some $x$ belonging to $[2, 3]$

(C) $g''(x) = 2$ for some $x$ belonging to $[1, 3]$

(D) $g''(x) = 2$ for some $x$ belonging to $[1.5, 2.5]$

Correct answer is (D).

I figured that $g(x) = x^2$, and therefore its second derivative must be $2$ in any interval. My understanding is clearly incorrect; please help me understand why $g''(x)=2$ is not necessarily true for all $x$. Am I missing something extremely obvious? I am so embarrassed!

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    If (D) is correct then so is (C) isn't it? – 2017-01-08
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    Just because $x^2$ fits the data doesn't mean that it is the only function that $g$ could be. There are an infinite number of other choices. Just among the polynomials, there are higher degree polynomials that would work. These would not have a constant second derivative. – 2017-01-08
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    Final clarification- when we are told a polynomial is $n$ times differentiable, does it mean the polynomial is exactly of degree $n$, or *at least* $n$? – 2017-01-08
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    By example, let $g(x)=x^2\cos(2\pi x)$, then $$g''(x)=-4 \pi ^2 x^2 \cos (2 \pi x)-8 \pi x \sin (2 \pi x)+2 \cos (2 \pi x)\neq 2$$ – 2017-01-08
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    @Andy : Every polynomial is infinitely times differentiable. Most of those derivatives will be zero. It makes thus no sense to talk about polynomials that are "only" $n$ times differentiable. - A piecewise cubic function can also be twice continuously differentiable. – 2017-01-09

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As you observed, the second power $x^2$ has the indicated function values. Using the difference $f(x)=g(x)-x^2$ you get a function with 3 roots. You can now apply Rolle's theorem twice (or thrice, depending on what you count) to find that $f''(x)=0$ at some point in the interval.

You can also consider $h(x)=f(x+1)-f(x)$ where $h'(x)=0$ implies $f'(x)=f'(x+1)$ at that point, which may give a little more control on the intermediate points generated by Rolle's theorem.

However, you can also piece together $C^2$ functions that are linear on $[1.5,2.5]$ with the given values.