= Solution
Each smooth segment follows <Newton's law of cooling>: $T'=-k(T-T_0)<0$ and $T''=k^2(T-T_0)>0$. Thus the graph decreases and is convex on both sides of $t_1$. It starts at $\alpha T_0$, falls to $\alpha T_0/2$, jumps upward by $\beta$, and then decreases asymptotically to $T_0$.
\b[The right-hand limit at the jump is above the original temperature], because $\beta>\alpha T_0/2$ implies $T(t_1+)>\alpha T_0$. The vertical dotted segment in the sketch denotes an instantaneous jump, not a continuous evolution through intermediate temperatures. The example uses $\alpha=4$, $\beta/T_0=3$; its qualitative shape holds for all the stipulated parameters.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ia/paper-2-cooling.png]
{title=Cooling interrupted by an instantaneous temperature increase}
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