Combined SAT scores in the United States could be approximated by
$T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038$ $( 0 \leq t \leq 23 )$
In the years 1967 - 1990. Here t is the number of years since 1967, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result. Round coordinates to two decimal places if necessary.

A)(11.75, 989.90), (26.90, 1007.45)
B) (19.33, 998.67)
C) (23, 1004.56)
D) (0, 0)
E) (0, 0), (23, 1004.56)

Question

Combined SAT scores in the United States could be approximated by 
 $T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038$ $( 0 \leq t \leq 23 )$  
In the years 1967 - 1990. Here t is the number of years since 1967, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result. Round coordinates to two decimal places if necessary.
 
A)(11.75, 989.90), (26.90, 1007.45)
B) (19.33, 998.67)
C) (23, 1004.56)
D) (0, 0)
E) (0, 0), (23, 1004.56)

Quizplus · Accepted Answer

To find the points of inflection, we need to find the second derivative of $T(t)$ and then set it equal to zero to solve for $t$. The second derivative of $T(t)$ is $T''(t) = -0.0303t + 1.1712$. Setting this equal to zero gives $t = 38.67$, which is not within the given range of $0 \leq t \leq 23$. However, since the question might involve a mistake in calculation or interpretation, the correct approach to finding inflection points involves finding where the second derivative changes sign, which typically involves solving $T''(t) = 0$ correctly within the given domain. Without the exact calculations provided in the options, it's likely there was a mistake in interpreting the correct method to find inflection points or in the calculation itself. Given the options and the typical process for finding inflection points, none of the provided options correctly follow from the given function and its domain, suggesting a misunderstanding in the question's solution process.

Combined SAT Scores in the United States Could Be Approximated T(t)=−0.0101t3+0.5856t2−9.6t+1038T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038T(t)=−0.0101t3+0.5856t2−9.6t+1038

Combined SAT Scores in the United States Could Be Approximated $T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038$