Suppose the resident population P(in millions) of the United States can be modeled by $P = 0.00000583 t ^ { 3 } + 0.005003 t ^ { 2 } + 0.13776 t + 4.658 , - 6 \leq t \leq 193$ , where $t = 0$ corresponds to 1800. Analytically find the minimum and maximum populations in the U.S. for $- 6 \leq t \leq 193$ .
A)The population is minimum at $t = - 6$ and maximum at $t = 0$ .
B)The population is minimum at $t = 0$ and maximum at $t = 193$ .
C)The population is minimum at $t = 193$ and maximum at $t = - 6$ .
D)The population is minimum at $t = - 6$ and maximum at $t = 193$ .
E)The population is minimum at $t = 0$ and maximum at $t = - 6$ .

Question

Quizplus · Accepted Answer

To find the minimum and maximum populations, we need to analyze the given cubic function for its behavior over the interval $[-6, 193]$. Given the coefficients of the cubic term (positive) and the nature of cubic functions, the population decreases as $t$ approaches the lower bound of the interval and increases towards the upper bound. Therefore, the population is minimum at $t = -6$ and maximum at $t = 193$, without needing to calculate derivatives, because the question implies a continuous increase over the given time period based on the structure of the polynomial.

Suppose the Resident Population P(in Millions) of the United States P=0.00000583t3+0.005003t2+0.13776t+4.658,−6≤t≤193P = 0.00000583 t ^ { 3 } + 0.005003 t ^ { 2 } + 0.13776 t + 4.658 , - 6 \leq t \leq 193P=0.00000583t3+0.005003t2+0.13776t+4.658,−6≤t≤193

Suppose the Resident Population P(in Millions) of the United States $P = 0.00000583 t ^ { 3 } + 0.005003 t ^ { 2 } + 0.13776 t + 4.658 , - 6 \leq t \leq 193$