Answer
(a) The surface area of a sphere as a function of radius is increasing function.
(b) Temperature at a point on the equator as a function of time is neither.
(c) Price of an airline ticket as a function of the price of oil is increasing function.
(d) Pressure of the gas in a piston as a function of volume is a decreasing function.
Work Step by Step
(a) The surface area of a sphere as a function of radius is given by $S(r)=4\pi r^2$. So, the first derivative is $S'(r)=8\pi r$ but since $r>0$, so, $S'(r)>0$, thus, by first derivative test the surface area is increasing.
(b) The temperature at a point can vary according to the position of earth in it's orbit which means the function will be periodic, i.e., repeats it's value after some time. thus, the temperature is neither.
(c) The price of an airline, $P$ is directly proportional to the cost of oil, $c$.
So, $P \propto c \Rightarrow P(c)=k\cdot c$, where $k$ is some positive constant, since, the price as well as cost both are positive quantities. Thus, the first derivative is $P'(c)=k>0$, which means price is increasing function.
(d) The pressure of the gas in a piston can be given by $P=\dfrac{nRT}{V}$ using ideal gas law, where $P, n, R, T$ and $V$ are pressure, number of moles, gas constant, temperature and volume, respectively.
Thus, the function of pressure is $P(V)=\dfrac{nRT}{V}$, which on differentiating gives
$P'(V)=\dfrac{-nRT}{V^2}<0$, so, by the first derivative test the pressure is decreasing function.
