1. **Question 1: Optimization Problem in Calculus**

   Consider a rectangular field with fixed perimeter P. Express the area A of the rectangle in terms of one of its side lengths (let's call it x). Formulate an optimization problem to find the maximum possible area for this rectangle. Utilize calculus to determine the critical points, analyze the second derivative test for concavity, and conclude whether the critical point yields a maximum or minimum area. Finally, interpret the results in the context of the original problem.

2. **Question 2: Differential Equations in Physics**

   A particle of mass m is subject to a resistive force directly proportional to its velocity v, given by the equation (F_{text{res}} = -bv), where b is a positive constant. The particle is also subjected to an external force (F_{text{ext}} = mg - kv), where g is the acceleration due to gravity and k is another positive constant. Formulate the second-order linear differential equation representing the motion of this particle. Solve the differential equation and interpret the physical significance of the constants and solutions in the context of the particle's behavior.

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