Suppose f''(x) exists for all real x. If f(2) | vedclass

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(B) Let $f(x)$ be a function such that $f(2)=2, f(3)=5, f(4)=10$. By Lagrange's Mean Value Theorem on $[2, 3]$,there exists $c_1 \in (2, 3)$ such that

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$f''(x) > 1$ for some $x \in (2, 4)$

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(B) Let $f(x)$ be a function such that $f(2)=2, f(3)=5, f(4)=10$. By Lagrange's Mean Value Theorem on $[2, 3]$,there exists $c_1 \in (2, 3)$ such that $f'(c_1) = \frac{f(3)-f(2)}{3-2} = \frac{5-2}{1} = 3$. By Lagrange's Mean Value Theorem on $[3, 4]$,there exists $c_2 \in (3, 4)$ such that $f'(c_2) = \frac{f(4)-f(3)}{4-3} = \frac{10-5}{1} = 5$. Now,applying the Mean Value Theorem to $f'(x)$ on the interval $[c_1, c_2]$,there exists $c \in (c_1, c_2) \subset (2, 4)$ such that $f''(c) = \frac{f'(c_2)-f'(c_1)}{c_2-c_1} = \frac{5-3}{c_2-c_1} = \frac{2}{c_2-c_1}$. Since $c_1 \in (2, 3)$ and $c_2 \in (3, 4)$,the length of the interval $c_2-c_1$ is less than $2$. Specifically,$0 Therefore,$f''(c) = \frac{2}{c_2-c_1} > \frac{2}{2} = 1$. Thus,$f''(x) > 1$ for some $x \in (2, 4)$.

Suppose $f''(x)$ exists for all real $x$. If $f(2) = 2$,$f(3) = 5$ and $f(4) = 10$,then which one among the following statements is definitely true?

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