A function r(t) gives the rate of change (in °C/min) of the temperature of a chemical solution during the interval 0≤t≤12 minutes. The graph of r is sketched on a grid and is described verbally below.
• From t=0 to t=3, r is positive and increases steadily.
• From t=3 to t=6, r remains positive but decreases to 0 by t=6.
• From t=6 to t=10, r is negative, reaching its minimum at t=8.
• From t=10 to t=12, r increases back to 0.
No numerical scale for r is provided; only the sign and general shape are known.
(a) Identify the subintervals of [0,12] on which the temperature of the solution is increasing and the subintervals on which it is decreasing. Justify your answer using the graphical information.
(b) Without performing any numerical calculation, determine whether the temperature at t=12 is warmer, cooler, or the same as the temperature at t=0. Explain your reasoning in terms of signed areas.
(c) The laboratory technician claims that the total area of the region above the t–axis is exactly equal in magnitude to the total area of the region below the axis. Explain what that claim would imply about the overall change in temperature on [0,12], and discuss whether that claim is consistent with the qualitative description of the graph.