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 (4, 101.8) b. the relative maximum point(s) is/are c. the relative minimum point(s) is/are and the relative maximum point(s) is/are . sketch a graph of the function. choose the correct graph below.

Explanation:

Step1: Identify the function

Let \(T(t)= - 0.2t^{2}+1.6t + 98.6\), where \(T\) is temperature and \(t\) is time.

Step2: Find the derivative

Using the power - rule \((x^n)^\prime=nx^{n - 1}\), \(T^\prime(t)=\frac{d}{dt}(-0.2t^{2}+1.6t + 98.6)=-0.4t + 1.6\).

Step3: Set the derivative equal to zero

Solve \(-0.4t + 1.6 = 0\).
\(-0.4t=-1.6\), so \(t = 4\).

Step4: Find the second - derivative

\(T^{\prime\prime}(t)=\frac{d}{dt}(-0.4t + 1.6)=-0.4<0\). Since \(T^{\prime\prime}(4)<0\), the function has a relative maximum at \(t = 4\).

Step5: Find the value of the function at \(t = 4\)

\(T(4)=-0.2\times4^{2}+1.6\times4 + 98.6=-0.2\times16 + 6.4+98.6=-3.2 + 6.4+98.6 = 101.8\). So the relative maximum point is \((4,101.8)\).

To sketch the graph, since the function \(T(t)=-0.2t^{2}+1.6t + 98.6\) is a quadratic function with \(a=-0.2<0\), the parabola opens downwards. The vertex is at \((4,101.8)\) and the \(y\) - intercept is at \(T(0)=98.6\).

Answer:

The relative maximum point is \((4,101.8)\)