Answer: D. $69$

Explanation:

$1$ . Interpret the function's behavior:

   - The function $f(x) = (x-a)(x-b)$ is a quadratic function whose roots (where $f(x) = 0$) are at $x=a$ and $x=b$.

   - We are told the function's value is positive at $x=32$ , negative at $x=35$, and positive again at $x=38$.

   - This means the graph of the parabola crosses the x-axis twice between $x=32$ and $x=38$.

$2$ . Locate the roots a and b:

   - Since $f(x)$ changes from positive to negative between $x=32$ and $x=35$, one root must be in this interval. So, $32 < a < 35$.

   - Since $f(x)$ changes from negative to positive between $x=35$ and $x=38$, the other root must be in this interval. So, $35 < b < 38$.

$3$ . Determine possible integer values for a and b:

   - Since a and b are integers, the only possible values are:

     - $a$ can be $33$ or $34$.

     - $b$ can be $36$ or $37$.

$4$ . Find the possible values for the sum $a+b$:

   - We can form pairs to find all possible sums:

     - If $a=33$ and $b=36$, then $a+b = 33+36=69$.

     - If $a=33$ and $b=37$, then $a+b = 33+37=70$.

     - If $a=34$ and $b=36$, then $a+b = 34+36=70$.

     - If $a=34$ and $b=37$, then $a+b = 34+37=71$.

   - The set of possible values for $a+b$ is $\{69, 70, 71\}$.

$5$ . Conclusion:

   - Of the choices given, only $69$ is in the set of possible values. This corresponds to option D.