1. **Problem statement:** Given the function $p(x) = g(x+1) - g(x)$ and $p(x) = 2$ for all $x$, determine which statements about $g$ must be true. 2. **Understanding the problem:** The function $p(x)$ represents the difference in $g$ values over an interval of length 1. Since $p(x) = 2$ is constant and positive, it means the change in $g$ over any interval of length 1 is always 2. 3. **Analyzing statement I:** "Because $p$ is positive and constant, the graph of $g$ always has positive slope." - The slope of $g$ at a point is the derivative $g'(x)$. - The difference quotient over interval length 1 is $p(x) = g(x+1) - g(x) = 2$. - The average rate of change over any interval of length 1 is 2, which is positive. - This suggests $g'(x)$ is positive everywhere if $g$ is differentiable. 4. **Analyzing statement II:** "Because $p$ is positive and constant, the graph $g$ is concave up." - Concavity relates to the second derivative $g''(x)$. - $p(x)$ being constant means the first difference is constant, but this does not imply the second difference (or second derivative) is positive. - $g$ could be linear (zero concavity) or have any concavity. - Therefore, statement II is not necessarily true. 5. **Analyzing statement III:** "Because $p$ is positive and constant, $g$ is increasing." - Since $p(x) = g(x+1) - g(x) = 2 > 0$, $g$ increases by 2 units for every increase of 1 in $x$. - This means $g$ is strictly increasing. 6. **Conclusion:** Statements I and III must be true, II does not have to be true. **Final answer:** (D) I and III only