Question

let ( f(t) ) be the temperature at time ( t ) where you live and suppose that at time ( t = 3 ) you feel uncomfortably hot. what happens to the temperature in each case?
(a) ( f(3)=2,f(3)= - 4 )
the temperature is increasing, and the rate of increase is increasing.
the temperature is increasing, but the rate of increase is decreasing.
the temperature is decreasing, but the rate of change is increasing (becoming less negative).
the temperature is decreasing, and the rate of change is decreasing (becoming more negative).
(b) ( f(3)=2,f(3)=4 )
the temperature is increasing, and the rate of increase is increasing.
the temperature is increasing, but the rate of increase is decreasing.
the temperature is decreasing, but the rate of change is increasing (becoming less negative).
the temperature is decreasing, and the rate of change is decreasing (becoming more negative).
(c) ( f(3)= - 2,f(3)= - 4 )
the temperature is increasing, and the rate of increase is increasing.
the temperature is increasing, but the rate of increase is decreasing.
the temperature is decreasing, but the rate of change is increasing (becoming less negative).
the temperature is decreasing, and the rate of change is decreasing (becoming more negative).
(d) ( f(3)= - 2,f(3)=4 )
the temperature is increasing, and the rate of increase is increasing.
the temperature is increasing, but the rate of increase is decreasing.
the temperature is decreasing, but the rate of change is increasing (becoming less negative).
the temperature is decreasing, and the rate of change is decreasing (becoming more negative).

Explanation:

Step1: Analyze the first - derivative

The first - derivative \(f^{\prime}(t)\) represents the rate of change of the temperature. If \(f^{\prime}(t)>0\), the temperature is increasing; if \(f^{\prime}(t)<0\), the temperature is decreasing.

Step2: Analyze the second - derivative

The second - derivative \(f^{\prime\prime}(t)\) represents the rate of change of the first - derivative. If \(f^{\prime\prime}(t)>0\), the rate of change of the temperature (given by \(f^{\prime}(t)\)) is increasing; if \(f^{\prime\prime}(t)<0\), the rate of change of the temperature (given by \(f^{\prime}(t)\)) is decreasing.

Case (a)

Since \(f^{\prime}(3) = 2>0\), the temperature is increasing. Since \(f^{\prime\prime}(3)=- 4<0\), the rate of increase (\(f^{\prime}(t)\)) is decreasing.

Case (b)

Since \(f^{\prime}(3) = 2>0\), the temperature is increasing. Since \(f^{\prime\prime}(3)=4>0\), the rate of increase (\(f^{\prime}(t)\)) is increasing.

Case (c)

Since \(f^{\prime}(3)=-2<0\), the temperature is decreasing. Since \(f^{\prime\prime}(3)=-4<0\), the rate of change (\(f^{\prime}(t)\)) is becoming more negative (decreasing).

Case (d)

Since \(f^{\prime}(3)=-2<0\), the temperature is decreasing. Since \(f^{\prime\prime}(3)=4>0\), the rate of change (\(f^{\prime}(t)\)) is becoming less negative (increasing).

Answer:

(a) The temperature is increasing, but the rate of increase is decreasing.
(b) The temperature is increasing, and the rate of increase is increasing.
(c) The temperature is decreasing, and the rate of change is decreasing (becoming more negative).
(d) The temperature is decreasing, but the rate of change is increasing (becoming less negative).