Microchip Transistor Thermal Cycle Profiling (Electrical & Computer Engineering)

In high-power embedded electronics, autonomous robotics controllers, and server power supplies, gallium-nitride (GaN) high-electron-mobility transistors (HEMTs) switch high currents at megahertz frequencies. Rapid pulse-width modulation (PWM) switching induces dynamic thermal dissipation across the silicon-substrate junction.

An electronic systems engineer tests the thermal profile of a GaN power transistor during a scheduled 6-second operational duty cycle. Over the closed test interval $t \in [0, 6]$ seconds, the internal junction temperature $T$ (in degrees Celsius, $^\circ\text{C}$) is accurately modeled by the cubic polynomial:

To ensure hardware reliability and prevent catastrophic component failure, the hardware design team must audit the thermal operating limits against strict safety specifications:

• The maximum junction temperature rating of the silicon package is $T_{\text{fail}} = 125^\circ\text{C}$. Exceeding this thermal ceiling triggers thermal runaway, melting internal bond wires.
• High-precision analog-to-digital signal calibration routines require minimal thermal noise interference, which occurs when the transistor is at its coolest operating point during the active cycle.

1. Locate all critical numbers of the junction temperature function $T(t)$ on the closed interval $[0, 6]$, and classify each critical number using the First Derivative Test.
2. Determine the absolute maximum temperature and absolute minimum temperature reached during the duty cycle, identifying both the extreme values and their exact timing locations.
3. Evaluate the component safety margin ($T_{\text{fail}} - T_{\max}$) and explain at what exact time instant firmware engineers should schedule precision sensor calibration to minimize thermal electronic noise.