The following is a True/False problem from an exam I got back:
If $f'(x)=1$ for each $x\in\mathbb{R}$ and $f(2)=7$ then $f(-2)=-7$.
I said false for this problem, but it was marked wrong. My reasoning for this was to treat this as an initial value problem. First, integrate $f'(x)$ so that $f(x)=\int f'(x)dx=x+c$. Next solve for $c$ from $f(2)=7$. Thus $2+c=7\implies c=5$. So our function is $f(x)=x+5$ where $f'(x)=1\forall x\in\mathbb{R}$ and $f(2)=7$. Both of the statements in the AND statement in the hypothesis are true, therefore the hypothesis is true. Next, checking $f(-2)=3\neq-7$ therefore the conclusion of the implication is false. Here we have True imply false, therefore the implication is false.
Is there an error in my rationale?
Your argument is correct. Besides, note that$$\frac{f(2)-f(-2)}{2-(-2)}=\frac{14}4=\frac72.$$But, by the mean value theorem, that quotient should be $1$.