Answer
Yes.
See proof below.
Work Step by Step
Let $f(x)=2x+5$.
Then, $f(0)=5$ and $f'(x)=2.$
Say that another function, $g(x)$ is such that $g(0)=2$ and $g'(x)=2$ for all x.
Then, by Corollary 2, since $f'(x)=g'(x)$ for every x, it follows that $g(x)=f(x)+C$ where $C$ is some constant.
We have:
$\left\{\begin{array}{ll}
g(0)=2, & f(0)=2\\
g(0)=f(0)+C &
\end{array}\right.\ \quad $
from which we find C:
$2=2+C$
$0=C$
So, $\quad g(x)=f(x).$
The answer is: yes.
