If $f(-10)=-5$ and $f^{\prime}(x) \geq 3$ for $x \in[-10,-5]$, what is the smallest value $f(-5)$ can possibly be?

Step 1 ：We are given that the derivative of the function \(f(x)\) is greater than or equal to 3 for \(x \in[-10,-5]\). This means that the function is increasing or constant in this interval.

Step 2 ：We also know that \(f(-10)=-5\).

Step 3 ：The smallest value \(f(-5)\) can possibly be is obtained when the function increases at the slowest rate, i.e., when the derivative is exactly 3.

Step 4 ：We can calculate this value by integrating the derivative from -10 to -5 and adding the initial value of the function.

Step 5 ：By doing this, we find that the smallest value \(f(-5)\) can possibly be is \(\boxed{10}\).