A chemical reaction takes place in a laboratory vessel, and its temperature T(t) (in degrees Celsius) at time t minutes is differentiable for all t≥0. The rate of change of temperature is modeled by
T′(t)=t2+1t2−4t.
At time t=0 minutes the temperature is 25∘C.
(a) Determine all critical numbers of T for t≥0.
(b) Using a sign chart or other analysis, classify each critical number as a time at which T attains a local maximum, a local minimum, or neither. Provide full justification.
(c) For t>4, indicate whether the temperature is increasing or decreasing and explain your reasoning based on the given model.
(d) Compute the temperature at every critical time and state the local maximum and/or local minimum temperatures of the reaction, including the corresponding times.