the temperature of a person during a certain illness is given by the following equation, where t is the…
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.
Answer
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))