Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 50

Suppose that $f(t)$ is the temperature at time $t$ where you live and when at $t = 3$ you feel uncomfortably hot. What will you feel on the following conditions?

$\begin{equation} \begin{aligned} \text{a.) } f'(3) &= 2, &&& f''(3) &= 4\\ \text{b.) } f'(3) &= 2, &&& f''(3) &= -4\\ \text{c.) } f'(3) &= -2, &&& f''(3) &= 4\\ \text{d.) } f'(3) &= -2, &&& f''(3) &= -4\\ \end{aligned} \end{equation}$


a.) Since $f'(3) > 0$, it means that the temperature is rising and if $f''(3) > 0$, it means that the temperature is not leveling off at any time. If I already feel hot, I might die on a heat stroke at some time.

b.) Since $f'(3) > 0$, it means that the temperature is rising. However, $f''(3) < 0$ it means that the temperature is about to level off. So I will feel hot for only a short period of time.

c.) Since $f'(3) < 0$, it means that the temperature is dropping and if $f''(3) > 0$, it means that the temperature is going to increase. So it looks like I will feel uncomfortable only for a short period of time.

d.) Since $f'(3) < 0$ and $f''(3) < 0$, the temperature is always decreasing. So I might suffer from Hypothermia at some time.