The fever hill

I had a cold, and there was a thermometer in the house. Every four hours someone wrote my temperature on a chart. This is one made-up example. When a fever needs a doctor is for a grown-up to decide.

A cold, and a thermometer. Every $4$ h my temperature went on a chart (a made-up example; a grown-up decides about a doctor).

$T(t)$ in degrees, every $4$ h (example).

At the start it was thirty-seven point zero degrees. Four hours later, thirty-eight point two. After eight hours, thirty-eight point nine: the top of the hill. I felt hot and heavy and wanted only to lie down.

Temperature $T$ in degrees, time $t$ in hours: $T(0)=37.0,T(4)=38.2,T(8)=38.9$. The top of the hill is $\max T=T(8)$.

$T(0)=37.0,T(4)=38.2,T(8)=38.9=\max T$.

Then the curve turned. After twelve hours, thirty-eight point four; after sixteen, thirty-seven point six; after twenty, thirty-six point nine. I could smell toast again.

Then: $T(12)=38.4,T(16)=37.6,T(20)=36.9$.

$T(12)=38.4,T(16)=37.6,T(20)=36.9$.

The chart went up by one point nine degrees in eight hours, and then down by two point zero in the next twelve. The surprise: I ended a tenth of a degree below where I began, as if the cold had lent me a little warmth and taken a little back as interest.

Up: $38.9-37.0=1.9$. Down: $38.9-36.9=2.0$. And $T(20)<T(0)$, by $37.0-36.9=0.1$.

$38.9-37.0=1.9$ and $38.9-36.9=2.0$, so $T(20)<T(0)$.