During a controlled fermentation process, the internal temperature T (in degrees Celsius) of a bioreactor is modeled for 0≤t≤10 hours by
T(t)=−0.04t3+0.6t2−2t+14.
At t=6 hours the batch is removed for sampling, and the reactor is temporarily shut off, causing the model to be undefined at t=6. After sampling, the reactor resumes operation and the same formula continues to model the temperature. Analyze the temperature extrema on the interval [0,10].
(a) Explain why the Extreme Value Theorem does not directly guarantee the existence of absolute extrema of T on [0,10].
(b) Determine all critical points of T on the two sub-intervals [0,6) and (6,10], showing all work.
(c) For each critical point found in part (b), classify it as a local maximum, a local minimum, or neither. Justify your classifications.
(d) Compute T at every critical point from part (c) and at each endpoint of the closed sub-intervals. Use these values to decide whether T attains an absolute maximum or absolute minimum on [0,10]. Discuss how the discontinuity at t=6 affects your conclusion.