The temperature of a person during a certain illness is given by the following equation, where \( T \) is the temperature \( \left({ }^{\circ} \mathrm{F}\right) \) at time \( t \), in days. Find the relative extreme points and sketch a graph of the function. \( T(t)=-03 t^{2}+24 t+983,0 \leq t \leq 8 \)

To find the relative extreme points of the function \( T(t) = -0.3t^2 + 24t + 983 \), we first need to find the derivative of the function and set it to zero to find critical points. ### Step 1: Find the derivative The derivative of \( T(t) \) is given by: \[ T'(t) = \frac{d}{dt}(-0.3t^2 + 24t + 983) = -0.6t + 24 \] ### Step 2: Set the derivative to zero To find the critical points, we set the derivative equal to zero: \[ -0.6t + 24 = 0 \] Solving for \( t \): \[ -0.6t = -24 \\ t = \frac{24}{0.6} = 40 \] ### Step 3: Check if the critical point is within the interval The critical point \( t = 40 \) is outside the interval \( [0, 8] \). Therefore, we need to evaluate the function at the endpoints of the interval to find the relative extreme points. ### Step 4: Evaluate \( T(t) \) at the endpoints 1. At \( t = 0 \): \[ T(0) = -0.3(0)^2 + 24(0) + 983 = 983 \] 2. At \( t = 8 \): \[ T(8) = -0.3(8)^2 + 24(8) + 983 \\ = -0.3(64) + 192 + 983 \\ = -19.2 + 192 + 983 \\ = 1155.8 \] ### Step 5: Determine the relative extreme points Since the critical point \( t = 40 \) is outside the interval, we only consider the values at the endpoints: - \( T(0) = 983 \) - \( T(8) = 1155.8 \) ### Conclusion - The minimum temperature occurs at \( t = 0 \) with \( T(0) = 983 \). - The maximum temperature occurs at \( t = 8 \) with \( T(8) = 1155.8 \). ### Step 6: Sketch the graph To sketch the graph of \( T(t) \): 1. The function is a downward-opening parabola (since the coefficient of \( t^2 \) is negative). 2. The vertex of the parabola is outside the interval, so the endpoints will give us the extreme values. 3. The graph will start at \( (0, 983) \) and end at \( (8, 1155.8) \). Here is a rough sketch of the graph: ``` T(t) | | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * |*_________________________ t 0 4 8 ``` The graph shows that the temperature increases from \( 983 \) at \( t = 0 \) to \( 1155.8 \) at \( t = 8 \).