### Instruction: Let $f$ be a linear function with the properties that $f(1) \le
f(2)$, $f(3) \ge f(4)$, and $f(5) = 5$.  Which of the following statements is true?

A) $f(0) < 0$

B) $f(0) = 0$

C) $f(1) < f(0) < f(-1)$

D) $f(0) = 5$

E) $f(0) > 5$.### Response: I know that a linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
I also know that the slope $m$ determines whether the function is increasing or decreasing as $x$ increases.
If $m > 0$, then the function is increasing, which means that $f(x)$ is larger for larger values of $x$.
If $m < 0$, then the function is decreasing, which means that $f(x)$ is smaller for larger values of $x$.
If $m = 0$, then the function is constant, which means that $f(x)$ is the same for any value of $x$.
Now, I can use the given properties of $f$ to figure out the sign of $m$.
Since $f(1) \le f(2)$, that means that $f$ is either increasing or constant from $x = 1$ to $x = 2$.
This implies that $m \ge 0$.
Since $f(3) \ge f(4)$, that means that $f$ is either decreasing or constant from $x = 3$ to $x = 4$.
This implies that $m \le 0$.
Combining these two inequalities, I get that $m = 0$.
This means that $f$ is a constant function, and its value is equal to $b$ for any $x$.
Since $f(5) = 5$, that means that $b = 5$.
Therefore, $f(x) = 5$ for any $x$, and in particular, $f(0) = 5$.
This matches option D.