Solve both parts and justify the conditions used. Part A: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique. Part B: A population is 800 and increases by 10% each year. Find it after two years.

1. Part A reasoning

2. Translate the context into an exponential or logarithmic equation and check what the model assumes.

3. Calculate or simplify this relation.

   $$f(1)=-1<0,\quad f(2)=1>0$$

4. A sign change alone does not establish uniqueness.

5. Part B reasoning

6. Translate the context into an exponential or logarithmic equation and check what the model assumes.

7. Calculate or simplify this relation.

   $$800(1.1)^2=968$$

8. This is compounding, not a single 20% increase.

A: Continuity gives existence; strict increase gives uniqueness. B: The requested value is 968.

Checks and common pitfalls: A sign change alone does not establish uniqueness. This is compounding, not a single 20% increase.