Question

the temperature of a person during a certain illness is given by the following equation, where t is the temperature (°f) at time t, in days. find the relative extreme points and sketch a graph of the function.
t(t)= - 0.2t² + 1.6t + 98.6, 0 ≤ t ≤ 8
what are the relative extreme points? select the correct choice below and fill in the answer box to complete your choice.
(simplify your answer. type an ordered - pair. use integers or decimals for any numbers in the expression. use a comma to separate answers as needed.)
a. the relative maximum point(s) is/are

b. the relative minimum point(s) is/are

c. the relative minimum point(s) is/are and the relative maximum point(s) is/are

Explanation:

Step1: Find the derivative

The function is $T(t)= - 0.2t^{2}+1.6t + 98.6$, and its derivative $T^\prime(t)$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $T^\prime(t)=-0.4t + 1.6$.

Step2: Set the derivative equal to zero

Set $T^\prime(t)=0$, so $-0.4t + 1.6 = 0$. Solving for $t$ gives $0.4t=1.6$, then $t = 4$.

Step3: Find the second - derivative

The second - derivative $T^{\prime\prime}(t)$ of $T(t)$ is $T^{\prime\prime}(t)=-0.4<0$. Since $T^{\prime\prime}(4)<0$, the function has a relative maximum at $t = 4$.

Step4: Find the value of the function at $t = 4$

Substitute $t = 4$ into $T(t)$: $T(4)=-0.2\times4^{2}+1.6\times4 + 98.6=-0.2\times16 + 6.4+98.6=-3.2 + 6.4+98.6=101.8$.

Answer:

A. The relative maximum point(s) is (4, 101.8)