Chemical Reactor Hot-Spot Optimization (Chemical Engineering)

Continuous temperature monitoring in exothermic reactors is critical for safety. The temperature profile across reaction zones is modeled by piecewise functions, which must be continuous and differentiable to avoid physical discontinuities in heat transfer models.

An exothermic chemical reactor has a temperature profile along its central axis modeled by the piecewise function:

where $x$ is the distance in meters from the inlet, and $T(x)$ is the temperature in degrees Celsius.

Determine the following:

1. Find the values of $a$ and $b$ such that the temperature profile is continuous and differentiable at the boundary $x = 3$ meters.
2. Optimize the reactor design by finding the maximum temperature along the axis in the interval $0 \le x \le 5$.