Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 5A / b / iii

Each smooth segment follows Newton's law of cooling: $T^{\prime }=-k(T-T_0)<0$ and $T^{\prime \prime }=k^2(T-T_0)>0$. 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 $T_0$.

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.